Approaching metric domains

APPR O ACHING METRIC DOMAINS GONC ¸ ALO GUTIERRES AN D DIRK HOFMA N N Abstra ct. In analogy to th e situation for contin u ous lattices which w ere introduced by Dana Scott as precisely the injectiv e T 0 spaces via the (no wada ys called) Scott topology , w e stu dy those metric s paces whic h corre sp ond to injec tive T 0 approac h spaces and c haracterise them as precisely the contin u ous lattices equipp ed with an unitary and associative [0 , ∞ ]-action. This result is achiev ed by a thorough analysis of the notion of cocompleteness for approach spaces. Introduction Domain theory is generally concerned with the study of or der e d sets admitting certain (typica lly up - directed) suprema and a notion of approximati on , here the latter amount s to sayi ng that eac h elemen t is a (up -directed) supremum of su itably deﬁn ed “ﬁnite” elemen ts. F rom a diﬀeren t p ersp ectiv e, domains can b e view ed as v ery particular top olo gic al sp ac e s ; in fact, in his pioneering pap er [Scott, 1972] Scott in tro duced the notion of con tinuous lattice p recisely as injectiv e top olog ical T 0 space. Y et another p oin t of view w as added in [Da y , 1975; Wyler, 1985] wh ere contin uous lattices are sho wn to b e precisely the algebr as for the ﬁlter monad. F urthermore, suitable su bmonads of the ﬁlter monad ha v e other typ es of d omains as algebras (for instance, conti nuous Scott domains [Gierz et al. , 2003] or Nac hbin’s ordered compact Hausd orﬀ spaces [Nac hbin, 1950]), and, as for con tinuous lattice s , these domains can b e equ ally seen as ob jects of top ology and of order th eory . This in terp la y b et w een top ology and algebra is v ery nicely explained in [Escard´ o and Flagg, 1999] where, emp lo ying a particular prop ert y of monads of ﬁlters, th e authors obtain “new pro ofs and [. . . ] new characte r izations of seman tic domains and top ological spaces b y injectivit y”. Since La wv ere ’s ground-breaking pap er [La wv ere , 1973] it is kno wn that an ind ividual metric sp aces X can b e view ed as a category with ob jects the p oin ts of X , an d the distance d ( x, y ) ∈ [0 , ∞ ] pla ys the role of th e “hom-set” of x and y . More mo dest, one can think of a metric d : X × X → [0 , ∞ ] as an order relation on X with truth-v alues in [0 , ∞ ] rather than in the Bo olean algebra 2 = { false , true } . In fact, writing 0 instead of true , > instead of ⇒ and additon + instead of and &, the reﬂ exivit y and transitivit y laws of an ordered set b ecome 0 ≥ d ( x, x ) and d ( x, y ) + d ( y , z ) ≥ d ( x, z ) ( x, y , x ∈ X ) , and in this p ap er we follo w La wv ere’s p oint of view and assume no f urther prop erties of d . As p oin ted out in [La wvere, 1973], “this connection is more fru itful than a mere analogy , b ecause it provides a sequence of mathematical theorems, so that enric hed category theo r y can suggest new d ir ections of researc h in metric space theory and conv er s ely”. A striking example of commonalit y b etw een category (resp. ord er ) 2010 Mathematics Subje ct Classiﬁc ation. 54A05, 54A20, 54E35, 54B30, 18B35. Key wor ds and phr ases. Contin uous lattice, metric space, ap p roac h space, injective space, co complete space. P artial ﬁnancial assistance by Cen tro de Matem´ atica d a Universidade de Coim bra/FCT , Centro de Inv estiga¸ c˜ ao e Desen- vol vimento em Matem´ atica e Aplica¸ c˜ oes da Universidade de Aveiro/F CT and the p ro ject MONDRIAN (u nder the contract PTDC/EIA-CCO/1083 02/2008) is gratefully ac kn o wledged. 1 2 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN theory and metric th eory wa s already giv en in [La w v ere , 1973] where it is shown that Cauc hy sequ ences corresp ond to adj oin t (bi)mo dules and con v ergence of Cauc hy sequences corresp onds to representabilt y of these mo d ules. Ev entually , this amoun ts to sa ying that a metric sp ace is Cauc hy complete if and only if it admits “suprema” of ce r tain “down-sets” (= morphisms of typ e X op → [0 , ∞ ]), here “suprema” has to b e tak en in the sense of w eight ed colimit of enr ic hed category theory [Eilen b erg and Kelly, 1966; Kelly, 1982]. Other t yp es of “do wn-sets” X op → [0 , ∞ ] sp ecify other prop erties of metric spaces: forw ard Cauc hy sequences (or nets) (see [Bonsangue et al. , 1998]) can b e represen ted b y so calle d ﬂ at mo dules (see [Vic kers, 2005]) and th eir limit p oints as “suprema” of these “do w n-sets”, an d the formal ball mo d el of a metric space relates to its cocompletion with r esp ect to y et another t yp e of “do wn-sets” (see [Rutten, 1998; Kostanek and W aszkiewicz, 2011]). The particular concern of this p ap er is to contribute to the d ev elopment of metric d omain theory . Due to the man y facets of domains, this can b e pursu ed b y either (1) form ulation order-theoretic concepts in the logic of [0 , ∞ ], (2) c on s idering injectiv e “[0 , ∞ ]-enric hed top ological spaces”, or (3) studying the algebras of “metric ﬁlter m onads”. Inspired by [La wv ere , 1973], there is a rich literature emp lo ying th e ﬁrst p oint of view, including W agner ’s Ph.D. thesis [W agner, 1994 ], the w ork of the Amsterdam researc h group at CWI [Bonsangue et al. , 1998; Rutten, 1998], the work of Flagg et al. on con tinuit y space s [Kopp erm an, 1988; Flagg, 1992, 1997b; Flagg et al. , 1996; F lagg and Kop p erman, 1997], and the w ork of W aszkiewicz with v arious coa u thors on app ro ximation and the formal ball mo del [W aszkiewicz, 2009; Hofmann and W aszkiewicz, 2011] and [Kostanek and W aszkiewicz, 2011]. Ho w eve r, in this pap er we tak e a diﬀerent approac h and concent rate on the second and third asp ect ab o ve. Our aim is to connect the theory of m etric spaces with the theory of “[0 , ∞ ]-enric hed top ologica l spaces” in a similar fashion as domain theory is supp orted b y top olog y , where b y “[0 , ∞ ]-enric hed top ological spaces” w e understand Lo we n’s approac h spaces [Lo wen, 1997]. (In a n u tshell, an approac h sp ace is to a topological sp ace what a metric space is to an ordered set: it can b e deﬁned in terms of ultraﬁlter con vergence where one associates to an ultraﬁlter x and a p oin t x a v alue of con ve r gence a ( x , x ) ∈ [0 , ∞ ] rather then ju st sa ying that x con verge s to x or not.) Th is idea was already pursu ed in [Hofmann, 2011] and [Hofmann, 2010 ] were among others it is sho wn that • injectiv e T 0 approac h spaces corresp ond bijectiv ely to a cla ss of metric spaces, h enceforth thought of as “con tinuous metric spaces”, • these “con tinuous metric s paces” are precisely the algebras for a certain monad on S et , henceforth though t of as the “metric ﬁlter monad”, • the category of injectiv e approac h spaces and approac h maps is C artesian closed. Here w e cont inue this path and • recal l the theory of metric and appr oac h spaces as generalised orders (resp. catego r ies), • c h aracterise m etric compact Hausdorﬀ spaces as the (suitably deﬁ ned) stably compact app roac h spaces, and • sho w that inj ective T 0 approac h sp aces (ak a “conti n u ous metric spaces”) can b e equiv alen tly describ ed as con tin u ous lattices equipp ed with an un itary and asso ciati ve action of the conti n u ous lattice [0 , ∞ ]. This result is ac h ieved by a thorough analysis of the notion of co completeness for approac h sp aces. Warning. The underlying ord er of a top ologi cal space X = ( X, O ) w e d eﬁne as x ≤ y whenev er  x → y APPRO ACHING METRIC DOMAINS 3 whic h is equiv alen t to O ( y ) ⊆ O ( x ), hence it is the dual of the sp e cialisation or der . As a consequence, th e underlying order of an inj ectiv e T 0 top ological space is the dual of a con tinuous lattice; and our results are stated in terms of these op-con tinuous lattices. W e h op e this d o es not create confu sion. 1. Metric sp aces 1.1. Preliminaries. According to the In tro duction, in this pap er w e consider metric sp ac es in a more general sense: a metric d : X × X → [0 , ∞ ] on set X is only required to satisfy 0 > d ( x, x ) and d ( x, y ) + d ( y , z ) > d ( x, z ) . F or con venience w e often also assume d to b e sep ar ate d meaning th at d ( x, y ) = 0 = d ( y, x ) implies x = y for all x, y ∈ X . With this n omenclature, “classical” metric spaces app ear now as separated, symmetric ( d ( x, y ) = d ( y , x )) and ﬁnitary ( d ( x, y ) < ∞ ) metric spaces. A map f : X → X ′ b et ween metric spaces X = ( X, d ) and X ′ = ( X ′ , d ′ ) is a metric map whenev er d ( x, y ) > d ′ ( f ( x ) , f ( y )) for all x, y ∈ X . The catego r y of metric spaces and metric maps w e denote as Met . T o ev ery metric space X = ( X , d ) on e asso ciates its dual sp ac e X op = ( X , d ◦ ) where d ◦ ( x, y ) = d ( y , x ), for all x, y ∈ X . Certainly , the metric d on X is symmetric if and only if X = X op . Ev ery metric map f b etw een metric spaces X and Y is also a metric map of t yp e X op → Y op , hen ce taking duals is actually a functor ( − ) op : Met → Met wh ic h sends f : X → Y to f op : X op → Y op . There is a canonical forgetful fu n ctor ( − ) p : Met → Ord : for a metric sp ace ( X, d ), pu t x ≤ y wh en ev er 0 > d ( x, y ) , and every metric map preserv es this order. Also note that ( − ) p : Met → Ord has a left adjoin t Ord → Met whic h in terpr ets an order relation ≤ on X as the metric d ( x, y ) = ( 0 if x ≤ y , ∞ else. In p articular, if X is a discrete ordered set meaning that the order relation is just the equalit y relation on X , th en one obtains the discrete metric on X wh ere d ( x, x ) = 0 and d ( x, y ) = ∞ for x 6 = y . The indu ced order of a metric space extends p oin t-wise to metric maps making Met an or der e d c ate gory , whic h enables us to talk ab out adjunctions . Here metric maps f : ( X , d ) → ( X ′ , d ′ ) and g : ( X ′ , d ′ ) → ( X, d ) form an adjun ction, written as f ⊣ g , if 1 X ≤ g · f and f · g ≤ 1 X ′ . Equiv alen tly , f ⊣ g if and only if, for all x ∈ X and x ′ ∈ X ′ , d ′ ( f ( x ) , x ′ ) = d ( x, g ( x ′ )) . The formula ab o v e explains the costume to call f left adjoin t and g right adjoin t. W e also recall that adjoin t maps determine eac h other meaning that f ⊣ g and f ⊣ g ′ imply g ≃ g ′ , and f ⊣ g and f ′ ⊣ g imply f ≃ f ′ . The category M et is complete and, for instance, the pr o duct X × Y of metric spaces X = ( X , a ) and ( Y , b ) is give n by the Cartesian pro du ct of the sets X and Y equipp ed with the max-metric d (( x, y ) , ( x ′ , y ′ )) = max( a ( x, x ′ ) , b ( y , y ′ )) . More int erestingly to us is the plus-metric d ′ (( x, y ) , ( x ′ , y ′ )) = a ( x, x ′ ) + b ( y , y ′ ) on the set X × Y , w e write a ⊕ b for this metric and denote the resulting metric space as X ⊕ Y . Note that the u nderlying order of X ⊕ Y is ju st the pro d uct order of X p and Y p in Ord . F urthermore, for metric maps f : X → Y and g : X ′ → Y ′ , the p ro duct of f and g giv es a metric map f ⊕ g : X ⊕ X ′ → Y ⊕ Y ′ , 4 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN and w e can view ⊕ as a fun ctor ⊕ : Met × Met → M et . This operation is b etter b eha ve d then the pro duct × in the sense that, for ev ery metric space X , the functor X ⊕ − : Met → Met has a righ t adjoin t ( − ) X : Met → M et send ing a metric space Y = ( Y , b ) to Y X = { h : X → Y | h in Met } with distance [ h, k ] = sup x ∈ X b ( h ( x ) , k ( x )) , and a metric m ap f : Y 1 → Y 2 to f X : Y X 1 → Y X 2 , h 7→ f · h. In particular, if X is a discrete space, then Y X is just the X -fold p o w er of Y . In the sequel we will pay particular atten tion to the metric sp ace [0 , ∞ ], with m etric µ d eﬁ ned by µ ( u, v ) = v ⊖ u := max { v − u, 0 } , for al l u, v ∈ [0 , ∞ ]. Then the un derlying ord er on [0 , ∞ ] is the “greater or equal relation” > . The “turning around” of the natural order of [0 , ∞ ] m igh t lo ok unmotiv ated at ﬁ rst sight but has its ro ots in the translation of “ fal se ≤ t rue ” in 2 to “ ∞ > 0” in [0 , ∞ ]. W e also note that u + − : [0 , ∞ ] → [0 , ∞ ] is left adjoin t to µ ( u, − ) : [0 , ∞ ] → [0 , ∞ ] with resp ect to > in [0 , ∞ ]. H ow ev er, in the sequel we will usually refer to th e natur al order 6 on [0 , ∞ ] with th e eﬀect that some formulas are dual to what one migh t exp ect. F or instance, the u n derlying monotone map of a metric map of t yp e X op → [0 , ∞ ] is of t yp e X p → [0 , ∞ ]; and w hen w e talk ab out a supr emum “ W ” in th e und erlying order of a generic metric space, it sp ecialises to taking in ﬁm u m “inf ” with resp ect to the usual ord er 6 on [0 , ∞ ]. F or ev ery set I , the maps inf : [0 , ∞ ] I → [0 , ∞ ] and sup : [0 , ∞ ] I → [0 , ∞ ] ϕ 7→ inf i ∈ I ϕ ( i ) ϕ 7→ sup i ∈ I ϕ ( i ) are metric maps, and so are + : [0 , ∞ ] ⊕ [0 , ∞ ] → [0 , ∞ ] and µ : [0 , ∞ ] op ⊕ [0 , ∞ ] → [0 , ∞ ] . ( u, v ) 7→ u + v ( u, v ) 7→ v ⊖ u More general, for a metric space X = ( X , d ), the metric d is a metric map d : X op ⊕ X → [0 , ∞ ]. Its mate is the Y one da emb e dding y X := p d q : X → [0 , ∞ ] X op , x 7→ d ( − , x ) , whic h satisﬁes indeed d ( x, y ) = [ y X ( x ) , y X ( y )] for all x, y ∈ X thanks to the Y oned a lemma wh ich states that [ y X ( x ) , ψ ] = ψ ( x ) , for all x ∈ X and ψ ∈ [0 , ∞ ] X op . 1.2. Co complete metric spaces. In this sub section w e h a ve a lo ok at metric sp aces “through the ey es of categ ory (resp. order) theory” and study the existence of sup r ema of “do wn-sets” in a metric space. This is a p articular case of the notion of w eigh ted colimit of enriched catego ries (see [Eilenberg and Kelly, 1966; K elly, 1982; Kelly and Schmitt, 2005], for in stance), and in this and the next s u bsection we sp ell out the meaning f or metric spaces of general notions and results of enr ic hed category theory . F or a metric space X = ( X , d ) and a “do wn-set” ψ : X op → [0 , ∞ ] in Met , an elemen t x 0 ∈ X is a supr emum of ψ whenever, for all x ∈ X , (1. i ) d ( x 0 , x ) = sup y ∈ X ( d ( y , x ) ⊖ ψ ( y )) . APPRO ACHING METRIC DOMAINS 5 Supr ema are un iqu e up to equiv alence ≃ , we write x 0 ≃ Sup X ( ψ ) and will frequently sa y the sup rem u m . F urtherm ore, a metric map f : ( X, d ) → ( X ′ , d ′ ) preserve s the su p rem u m of ψ ∈ [0 , ∞ ] X op whenev er d ′ ( f (Sup X ( ψ )) , x ′ ) = su p x ∈ X ( d ′ ( f ( x ) , x ′ ) ⊖ ψ ( x )) for all x ′ ∈ X ′ . As for ordered sets: Lemma 1.1. L eft adjoint metric maps pr eserve al l supr e ma. A metric sp ace X = ( X , d ) is called c o c omplete if ev ery “do wn -set” ψ : X op → [0 , ∞ ] has a supremum. This is th e case pr ecisely if, for all ψ ∈ [0 , ∞ ] X op and all x ∈ X , d (Sup X ( ψ ) , x ) = sup y ∈ X ( d ( y , x ) ⊖ ψ ( y )) = [ ψ , y X ( x )]; hence X is co complete if an d only if the Y oned a emb ed ding y X : X → [0 , ∞ ] X op has a left adjoin t Sup X : [0 , ∞ ] X op → X in Met . More generally , one has (see [Hofmann , 2011], for instance) Prop osition 1.2. F or a metric sp ac e X , the fol lowing c onditions ar e e qui valent. (i) X i s i nje ctiv e (with r esp e c t to isometries). (ii) y X : X → [0 , ∞ ] X op has a left inverse. (iii) y X has a left adjoint. (iv) X is c o c omplete. Here a metric map i : ( A, d ) → ( B , d ′ ) is called isometry if one has d ( x, y ) = d ′ ( i ( x ) , i ( y )) for all x, y ∈ A , and X is injectiv e if, for all isometries i : A → B and all f : A → X in Met , there exists a metric map g : B → X with g · i ≃ f . Dually , an in ﬁm um of an “up-set” ϕ : X → [0 , ∞ ] in X = ( X , d ) is an element x 0 ∈ X s uc h that, for all x ∈ X , d ( x, x 0 ) = s up y ∈ X ( d ( x, y ) ⊖ ϕ ( y )) . A metric space X is c omplete if ev ery “up-set” has an inﬁ m u m. By deﬁnition, an inﬁmum of ϕ : X → [0 , ∞ ] in X is a supr emum of ϕ : ( X op ) op → [0 , ∞ ] in X op , and eve r ything said ab ov e can b e r ep eated no w in its dual form. In p articular, with h X : X →  [0 , ∞ ] X  op , x 7→ d ( x, − ) denoting the c ontr avariant Y one da emb e dding (whic h is th e dual of y X op : X op → [0 , ∞ ] X ): Prop osition 1.3. F or a metric sp ac e X , the fol lowing c onditions ar e e qui valent. (i) X i s i nje ctiv e (with r esp e c t to isometries). (ii) h X : X →  [0 , ∞ ] X  op has a left inverse. (iii) h X has a left adjoint. (iv) X is c omplete. Corollary 1.4. A metric sp ac e i s c omplete if and only if it is c o c omplete. This latter fact can b e also seen in a d iﬀeren t wa y . T o ev ery “do wn-set” ψ : X op → [0 , ∞ ] one assigns its “up-set of u pp er b ound s” ψ + : X → [0 , ∞ ] , x 7→ sup y ∈ X ( d ( y , x ) ⊖ ψ ( y )) , and to every “up -set” ϕ : X → [0 , ∞ ] its “down-set of lo wer b ounds ” ϕ − : X op → [0 , ∞ ] , x 7→ s u p y ∈ X ( d ( x, y ) ⊖ ϕ ( y )) . 6 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN This w ay one deﬁn es an adjun ction ( − ) + ⊣ ( − ) −  [0 , ∞ ] X  op ( − ) − - - ⊤ [0 , ∞ ] X op ( − ) + m m X h X e e J J J J J J J J J J y X ; ; v v v v v v v v v in M et w h ere b oth maps comm ute with the Y oneda embedd ings. Therefore a left inv erse of y X pro du ces a left inv erse of h X , and vice versa. Th e adju nction ( − ) + ⊣ ( − ) − is also know as the Isb el l c onjugation adjunction . F or ev ery (co)co mplete metric space X = ( X, d ), its underlyin g ordered set X p is (co) complete as w ell. T his follo ws for instance from the f act that ( − ) p : Met → Ord preserves injectiv e ob jects. Another argumen t go es as follo w s. F or ev ery (down-set) A ⊆ X , one deﬁnes a m etric m ap ψ A : X op → [0 , ∞ ] , x 7→ inf a ∈ A d ( x, a ) , and a supremum x 0 of ψ A m u s t satisfy , for all x ∈ X , d ( x 0 , x ) = sup y ∈ X ( d ( y , x ) ⊖ ψ A ( y )) = sup a ∈ A sup y ∈ X ( d ( y , x ) ⊖ d ( y , x )) = sup a ∈ A d ( a, x ) . Therefore x 0 is not only a supremum of A in the ordered set X p , it is also preserve d by every monotone map d ( − , x ) : X p → [0 , ∞ ]. Lemma 1.5. L et X = ( X, d ) b e a metric sp ac e and let x 0 ∈ X and A ⊆ X . Then x 0 is the supr emum of ψ A if and only if x 0 is the (or der the or etic) supr emum of A and, f or every x ∈ X , the monotone map d ( − , x ) : X p → [0 , ∞ ] pr eserve s this supr emum. 1.3. T ensored metric spaces. W e are n o w intereste d in those metric sp aces X = ( X , d ) whic h admit suprema of “down-sets” of the form ψ = d ( − , x ) + u where x ∈ X and u ∈ [0 , ∞ ]. In the sequel w e write x + u in stead of Sup X ( ψ ). According to (1. i ), the elemen t x + u ∈ X is characte r ised up to equiv alence b y d ( x + u, y ) = d ( x, y ) ⊖ u, for all y ∈ X . A metric m ap f : ( X, d ) → ( X ′ , d ′ ) pr eserv es the supremum of ψ = d ( − , x ) + u if and only if f ( x + u ) ≃ f ( x ) + u . Dually , an inﬁmum of an “up -set” of the form ϕ = d ( x, − ) + u we d enote as x ⊖ u , it is c haracterised u p to equiv alence by d ( y , x ⊖ u ) = d ( y , x ) ⊖ u. One calls a m etric space tensor e d if it admits all sup r ema x + u , and c otensor e d if X admits all inﬁ ma x ⊖ u . Example 1.6. T he m etric s p ace [0 , ∞ ] is tensored and cotensored where x + u is giv en by add ition and x ⊖ u = max { x − u, 0 } . Note that X = ( X , d ) is tens ored if and only if every d ( x, − ) : X → [0 , ∞ ] has a left adjoint x + ( − ) : [0 , ∞ ] → X in Met , and X is cotensored if and only if every d ( − , x ) : X op → [0 , ∞ ] has a left adjoint x ⊖ ( − ) : [0 , ∞ ] → X op in Met . F ur thermore, if X is tensored and cotensored, then ( − ) + u : X → X is left adjoint to ( − ) ⊖ u : X → X in Met , f or eve r y u ∈ [0 , ∞ ]. Theorem 1.7. L et X = ( X, d ) b e a metric sp ac e. Then the fol lowing assertions ar e e quiv alent. (i) X i s c o c omplete. (ii) X has al l or der-the or etic supr ema and is tensor e d and c otensor e d. APPRO ACHING METRIC DOMAINS 7 (iii) X has al l (or der th e or etic) supr ema, is tensor e d and, for every u ∈ [0 , ∞ ] , the monotone map ( − ) + u : X p → X p has a right adjoint in Ord . (iv) X has al l (or der the or etic) supr ema, is tenso r e d and, for every u ∈ [0 , ∞ ] , the monotone ma p ( − ) + u : X p → X p pr eserves su pr ema. (v) X has al l (or der the or etic) supr ema, is tensor e d and, for every x ∈ X , the monoto ne map d ( − , x ) : X p → [0 , ∞ ] has a right adjoint in Ord . (vi) X is has al l (or der the or etic) supr ema, is tensor e d and, for every x ∈ X , the monot one map d ( − , x ) : X p → [0 , ∞ ] pr eserves supr ema. Under these c onditions, the supr emum of a “down-set” ψ : X op → [0 , ∞ ] c an b e c alculate d as (1. ii ) Sup ψ = inf x ∈ X ( x + ψ ( x )) . A metric map f : X → Y b etwe en c o c omplete metric sp ac es pr e serves al l c olimits if and only if it pr eserve s tensors and supr ema. Pr o of. By the pr eceding discus sion, the implications (i) ⇒ (ii) and (ii) ⇒ (iii) are obvi ous, and so are (iii) ⇔ (iv) and (v) ⇔ (vi). T o see (iii) ⇒ (v), just note th at a r igh t adjoin t ( − ) ⊖ u : X p → X p of ( − ) + u pro du ces a r igh t adjoin t x ⊖ ( − ) : [0 , ∞ ] → X p of d ( − , x ). Finally , (vi) ⇒ (i) can b e shown b y ve r ifying that (1. ii ) calculates in d eed a supr em um of ψ .  Ev ery m etric s p ace X = ( X , d ) induces metric maps X ⊕ [0 , ∞ ] B X − − → [0 , ∞ ] X op and X I F X,I − − − → [0 , ∞ ] X op (where I is an y set) . Here B X : X ⊕ [0 , ∞ ] → [0 , ∞ ] X op , ( x, u ) 7→ d ( − , x ) + u is the mate of th e comp osite X op ⊕ X ⊕ [0 , ∞ ] d ⊕ 1 − − − → [0 , ∞ ] ⊕ [0 , ∞ ] + − → [0 , ∞ ] , and F X,I : X I → [0 , ∞ ] X op is the m ate of the comp osite X op ⊕ X I → [0 , ∞ ] I inf − − → [0 , ∞ ] , where the ﬁrst comp onent is the mate of the comp osite X op ⊕ X I ⊕ I 1 ⊕ ev − − − → X op ⊕ X d − → [0 , ∞ ] . Sp elled out, for ϕ ∈ X I and x ∈ X , F X,I ( ϕ )( x ) = inf i ∈ I d ( x, ϕ ( i )), and a suprem u m of F X,I ( ϕ ) ∈ [0 , ∞ ] X op is also a (order-theoretic) supr em um of the family ( ϕ ( i )) i ∈ I in X . If X is cocomplete, by comp osing with Sup X one obtains metric maps X ⊕ [0 , ∞ ] + − → X and X I W − → X (where I is an y set) . Finally , a catego rical standart argumen t (see [Johnstone, 1986, Lemma 4.10] sho ws that with Y also Y X is injectiv e, h ence, Y X is cocomplete. F ur thermore, tensors and suprema in Y X can b e calcula ted p oint wise: h + u = ( − + u ) · h and _ i ∈ I h i ! = _ ·h h i i i ∈ I , for u ∈ [0 , ∞ ], h ∈ Y X and h i ∈ Y X ( i ∈ I ). Here h h i i i ∈ I : X → Y I denotes the map indu ced b y the family ( h i ) i ∈ I . 8 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN 1.4. [0 , ∞ ] -actions on ordered sets. When X = ( X, d ) is a tensored metric space, w e m ight not hav e Sup X deﬁned on the whole s pace [0 , ∞ ] X op , but it still is deﬁned on its subspace of all metric maps ψ : X op → [0 , ∞ ] of the form ψ = d ( − , x ) + u . Hence, one still has a metric map X ⊕ [0 , ∞ ] → X , ( x, u ) 7→ x + u, and one easily veriﬁes the f ollo w ing pr op erties. (1) F or all x ∈ X , x + 0 ≃ x . (2) F or all x ∈ X and all u, v ∈ [0 , ∞ ], ( x + u ) + v ≃ x + ( u + v ). (3) + : X p × [0 , ∞ ] → X p is monotone in the ﬁrs t and an ti-monotone in the second v ariable. (4) F or all x ∈ X , x + ∞ is a b ottom elemen t of X p . (5) F or all x ∈ X and ( u i ) i ∈ I in [0 , ∞ ], x + inf i ∈ I u i ≃ _ i ∈ I ( x + u i ). Of cours e, (4) is a sp ecial case of (5). If X is separated, then the ﬁrst thr ee conditions just tell us that X p is an algebra for the monad induced by the monoid ([0 , ∞ ] , > , + , 0) on Ord sep . Hence, X 7→ X p deﬁnes a forgetful functor Met sep , + → Ord [0 , ∞ ] sep , where Met sep , + denotes the category of tensored and separated metric spaces and tensor p reserving met- ric maps, and Ord [0 , ∞ ] sep the category of separated ordered sets with an unitary (i.e. s atisfying (1)) and asso ciativ e (i.e. satisfying (2)) action of ([0 , ∞ ] , > , + , 0) ([0 , ∞ ]-algebras for sh ort) and monotone maps whic h preserve this action. Con versely , let no w X b e a [0 , ∞ ]-algebra with action + : X × [0 , ∞ ] → X . W e deﬁn e (1. iii ) d ( x, y ) = inf { u ∈ [0 , ∞ ] | x + u ≤ y } . Certainly , x ≤ y imp lies 0 > d ( x, y ), in particular one has 0 > d ( x, x ). Since, for x, y , z ∈ X , d ( x, y ) + d ( y , z ) = in f { u ∈ [0 , ∞ ] | x + u ≤ y } + inf { v ∈ [0 , ∞ ] | y + v ≤ z } = inf { u + v | u, v ∈ [0 , ∞ ] , x + u ≤ y , y + v ≤ z } > inf { w ∈ [0 , ∞ ] | x + w ≤ z = d ( x, z ) , w e ha v e seen that ( X , d ) is a metric space. If the [0 , ∞ ]-algebra X comes from a tensored separated metric space, then we get the original metric bac k. If X satisﬁes (4), then the inﬁmum in (1. iii ) is non-empt y , and therefore d ( x + u, y ) = inf { v ∈ [0 , ∞ ] | x + u + v ≤ y } = inf { w ⊖ u | w ∈ [0 , ∞ ] , x + w ≤ y } = inf { w | w ∈ [0 , ∞ ] , x + w ≤ y } ⊖ u = d ( x, y ) ⊖ u, hence ( X , d ) is tensored wh ere + is giv en b y the algebra op eration. Finally , if X satisﬁes (5), then the inﬁmum in (1. iii ) is actually a m inim u m, and therefore 0 > d ( x, y ) imp lies x ≤ y . All told: Theorem 1.8. The c ate gory Met sep , + is e q u ivalent to th e ful l sub c ate gory of Ord [0 , ∞ ] sep deﬁne d by those [0 , ∞ ] -algebr as satisfying (5). Under this c orr esp ondenc e , ( X, d ) is a c o c omplete sep ar ate d metr i c sp ac e if and only i f th e [0 , ∞ ] -algebr a X has al l supr ema and ( − ) + u : X → X pr eserves supr ema, for al l u ∈ [0 , ∞ ] . R emark 1.9 . The second p art of th e theorem ab o ve is essent ially in [P edicchio and Tholen, 1989], which actually states that co complete separated metric spaces c orresp ond precisely to su p-lattices equipp ed with an unitary and asso ciativ e action + : X × [0 , ∞ ] → X whic h is a bimorphism , m eaning that it APPRO ACHING METRIC DOMAINS 9 preserve s suprema in eac h v ariable (where the order on [0 , ∞ ] is > ) but not necessarily in b oth. Thanks to F reyd ’s Adj oin t F unctor Theorem (see [MacLane , 1971, Section V.6]), the category Sup of sup-lattices and suprema preserving m aps adm its a tensor p r o duct X ⊗ Y wh ich is c h aracterised by Bimorph( X × Y , Z ) ≃ Sup ( X ⊗ Y , Z ) , naturally in Z ∈ Sup , for all sup-latt ices X , Y . Hence, a co complete separated metric space can b e iden tiﬁed with a su p-lattice X equipp ed with an unitary and asso ciativ e actio n + : X ⊗ [0 , ∞ ] → X in Sup . Prop osition 1.10. L et X and Y b e tensor e d metric sp ac es and f : X → Y b e a map. Then f : X → Y is a metric map if and only if f : X p → Y p is monotone and, for al l x ∈ X and u ∈ [0 , ∞ ] , f ( x )+ u ≤ f ( x + u ) . Pr o of. Ev er y metric map is also mon otone with resp ect to the und erlying orders and satisﬁes f ( x ) + u ≤ f ( x + u ), for all x ∈ X and u ∈ [0 , ∞ ]. T o see the rev er s e imp lication, recall that the metric d on X satisﬁes d ( x, y ) = inf { u ∈ [0 , ∞ ] | x + u ≤ y } , and for th e m etric d ′ on Y one has d ( f ( x ) , f ( y )) = in f { v ∈ [0 , ∞ ] | f ( x ) + v ≤ f ( y ) } . If x + u ≤ y , then f ( x ) + u ≤ f ( x + u ) ≤ f ( y ), and the assertion follo ws.  2. Metric comp act Hausdorff sp aces and ap p ro ach sp aces 2.1. Con tinuous lattices. Con tinuous lattice s w ere int ro duced by D. Scott [Scott, 1972] as p recisely those orders app earing as the sp e cialisation or der of an injectiv e top olog ical T 0 -space. Here, for an arbitrary top ological space X with top ology O , the sp ecial isation order ≤ on X is deﬁn ed as x ≤ y when ever O ( x ) ⊆ O ( y ) , for all x, y ∈ X . This relation is alw a ys reﬂexive and tr ansitiv e, and it is ant i-symmetric if and only if X is T 0 . I f X is an inj ective T 0 -space, then the ord ered s et ( X , ≤ ) is actually complete and , f or all x ∈ X , x = _ { y ∈ X | y ≪ x } (where y ≪ x wh enev er x ≤ W D ⇒ y ∈ D for ev ery up-directed d own-set D ⊆ X ); in general, a complete separated ordered set with this prop ert y is called c ontinuous lattic e . In this particular case the sp ecialisation order con tains all in formation ab out th e top ology of X : A ⊆ X is op en if an d only if A is unreac h able b y up-d irected suprema in the sense that (2. i ) _ D ∈ A ⇒ D ∩ A 6 = ∅ for ev ery up-directed do wn -set D ⊆ A . Quite generally , (2. i ) deﬁnes a top ology on X for any ordered set X , and a monotone map f : X → Y is con tinuous with resp ect to these top ologies if and only if f preserve s all existing up-directed suprema. F urthermore, th e sp ecialisat ion order of th is top ology gives the original order bac k, and one obtai ns an injectiv e top ological T 0 -space if and only if X is a con tinuous lattice . In the sequel we will consider topological spaces mostly via ultraﬁlter co n vergence, and therefore d eﬁ ne the underlying or der ≤ of a top ological space as the “p oin t shadow” of this con vergence : x ≤ y when ever  x → y , (  x = { A ⊆ X | x ∈ A } ) 10 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN whic h is dual to the sp ecialisa tion order . Consequentl y , the under lyin g order of an injectiv e top ological T 0 space is an op-c ontinuous lattic e meaning that ( X, ≤ ) is complete and, for an y x ∈ X , x = ^ { y ∈ X | y ≻ x } , where y ≻ x w henev er x ≤ V D implies y ∈ D for every do wn-dir ected up-set D ⊆ X . W e also note that, with resp ect to th e underlying order, th e conv ergence relation of an injectiv e T 0 space is giv en by x → x ⇐ ⇒   ^ A ∈ x _ y ∈ A y   ≤ x, for all ultraﬁlters x on X and all x ∈ X . 2.2. Ordered compact Hausdorﬀ spaces. The class of domains with arguably the most d ir ect general- isation to metric spaces is th at of stably compact spaces, or equiv alently , or der e d c omp act Hausdorﬀ sp ac e s . The latter w er e introdu ced b y [Nac h bin, 1950] as triples ( X, ≤ , O ) w here ( X , ≤ ) is an ordered set (we do not assume an ti-symmetry here) and O is a compact Hausd orﬀ top olog y on X so that { ( x, y ) | x ≤ y } is closed in X × X . A morp hism of ordered compact Hausd orﬀ spaces is a map f : X → Y which is b oth monotone and con tinuous. T he resulting catego r y of ordered compact Hausdorﬀ space and morphisms w e denote as OrdCompHaus . If ( X , ≤ , O ) is an ord er ed compact Hausdorﬀ spaces, then the dual order ≤ ◦ on X together w ith the top ology O d eﬁnes an ordered compact Hausdorﬀ s paces ( X, ≤ ◦ , O ), and on e obtains a functor ( − ) op : OrdCompHaus → OrdCompHaus wh ic h comm utes with the canonical forgetful functor OrdCompHaus → Set . Analogously to the fact that compact Hausdorﬀ spaces and cont inuous maps form an algebraic category o v er S et v ia ultraﬁlter conv ergence U X → X [Manes, 1969], it is shown in [Flagg, 1997a] that the full sub category OrdCompHaus sep of O rdCompHaus deﬁn ed b y those spaces with anti-symmetric order is the catego r y of Eilenberg-Mo ore algebras for the prime ﬁ lter monad of up -sets on Ord sep . The sit uation do es not change m uc h when w e drop an ti-symmetry , in [Tholen, 2009] it is sho wn that ordered compact Hausdorﬀ sp aces are precisely the Eilen b erg-Moore algebras for the ultraﬁlter monad U = ( U, e, m ) suitably deﬁn ed on Ord . He r e the functor U : Ord → Ord sends an ordered set X = ( X , ≤ ) to the set U X of all u ltraﬁlters on the set X equipp ed with the order relation x ≤ y whenev er ∀ A ∈ x , B ∈ y ∃ x ∈ A, y ∈ B . x ≤ y ; ( x , y ∈ U X ) and the maps e X : X → U X m X : U U X → U X x 7→  x := { A ⊆ X | x ∈ A } X 7→ { A ⊆ X | A # ∈ X } (where A # := { x ∈ U X | A ∈ x } ) are monotone with resp ect to th is order relation. Then , for α : U X → X denoting the con v ergence of the compact Hausdorﬀ top ology O , ( X , ≤ , O ) is an ordered compact Hausdorﬀ space if and only if α : U ( X , ≤ ) → ( X , ≤ ) is monotone. 2.3. Metric compact H ausdorﬀ spaces. Th e presen tation in [Th olen, 2009] is ev en more general and giv es also an extension of the u ltraﬁ lter monad U to Met . F or a metric space X = ( X , d ) and ultraﬁlters x , y ∈ U X , one deﬁnes a distance U d ( x , y ) = sup A ∈ x ,B ∈ y inf x ∈ A,y ∈ B d ( x, y ) and tu r ns this w ay U X into a metric space. Th en e X : X → U X and m X : U U X → U X are metric maps and U f : U X → U Y is a m etric map if f : X → Y is so. Not surp risingly , we call an Eilen b erg–Moore algebra for this monad metric c omp act Hausdorﬀ sp ac e . Su c h a s p ace can b e describ ed as a triple ( X, d, α ) APPRO ACHING METRIC DOMAINS 11 where ( X, d ) is a metric s pace and α is (the con ve r gence relation of ) a compact Hausdorﬀ top ology on X so th at α : U ( X, d ) → ( X, d ) is a metric m ap. W e denote the categ ory of metric compact Hausdorﬀ spaces and m orphisms (i.e. maps whic h are b oth metric maps and con tin u ous) as MetCompHaus . The op eration “taking the dual metric space” lifts to an endofu nctor ( − ) op : MetCompHaus → MetCompHaus where X op := ( X, d ◦ , α ), for every metric compact Hausdorﬀ space X = ( X, d, α ). Example 2.1. The metric space [0 , ∞ ] with metric µ ( u, v ) = v ⊖ u b ecomes a metric compact Hausd orﬀ space with th e Euclidean compact Hausd orﬀ top ology w hose conv ergence is giv en by ξ ( v ) = sup A ∈ v inf v ∈ A v , for v ∈ U [0 , ∞ ]. Consequently , [0 , ∞ ] op denotes the metric compact Hausd orﬀ space ([0 , ∞ ] , µ ◦ , ξ ) with the same compact Hausd orﬀ top olog y on [0 , ∞ ] and with the metric µ ◦ ( u, v ) = u ⊖ v . Lemma 2.2. If ( X , d ) is a tensor e d metric sp ac e, then ( U X, U d ) is tensor e d to o. Pr o of. F or u ∈ [0 , ∞ ] and x ∈ U X , put x + u = U ( t u )( x ) wher e t u : X → X sen d s x ∈ X to (a c hoice of ) x + u . Then U d ( x + u, y ) = sup A ∈ x ,B ∈ y inf x ∈ A,y ∈ B d ( x + u, y ) = sup A ∈ x ,B ∈ y inf x ∈ A,y ∈ B d ( x, y ) ! ⊖ u = U d ( x , y ) ⊖ u, for all y ∈ U X . Here w e use the fact that − ⊖ u : [0 , ∞ ] → [0 , ∞ ] p reserv es all supr ema and n on -emp t y inﬁma.  Clearly , if f : X → Y is a tensor p reserving map b etw een tensored metric spaces, then U f ( x + u ) ≃ U f ( x ) + u , hence U : Met → Met r estricts to an endofunctor on the catego ry Met + of tensored metric spaces and tensor p reserving m aps. 2.4. Stably compact top ological spaces. As we ha v e alrea d y indicated at the b eginning of Su bsection 2.2, (anti- s y m metric) ordered compact Hausd orﬀ sp aces can b e equiv alen tly seen as sp ecial top ological spaces. In fact, b oth structures of an ord er ed compact Hausdorﬀ space X = ( X, ≤ , O ) can b e com bined to form a top ology on X whose op ens are p recisely those elemen ts of O whic h are do wn-sets in ( X , ≤ ), and th is pro cedure d eﬁnes indeed a functor K : OrdCompHaus → T op . An u ltraﬁ lter x ∈ U X con verges to a p oint x ∈ X with resp ect to this new top ology if and only if α ( x ) ≤ x , wh ere α : U X → X d enotes the con ve r gence of ( X , O ). Hence, ≤ is just the un d erlying order of O and α ( x ) is a smallest con vergence p oin t of x ∈ U X with resp ect to this order. F r om that it follo ws at once th at we can reco ver b oth ≤ and α from O . T o b e rigorous, th is is true when ( X , ≤ ) is an ti-symmetric, in the general case α is determined only up to equiv alence. In any case, w e deﬁn e the d ual of a top ological space Y of the form Y = K ( X, ≤ , α ) as Y op = K ( X, ≤ ◦ , α ), and note that equiv alen t m aps α lead to the same space Y op . A T 0 space X = ( X , O ) comes from a anti-symmetric ord ered compact Hausdorﬀ space precisely if X is stably c omp act , th at is, X is sob er, lo cally compact and stable. T he latter prop ert y can b e deﬁned in diﬀeren t manners, we use here the one giv en in [S immons, 1982]: X is stable if, for op en subsets U 1 , . . . , U n and V 1 , . . . , V n ( n ∈ N ) of X with U i ≪ V i for eac h 1 ≤ i ≤ n , also T i U i ≪ T i V i . As usu al, it is enough to require stabilit y un der empty and binary inte rsections, and stabilit y under empt y in tersection translates to compactness of X . Also note that a T 0 space is lo cally compact if and only it is exp onentia b le in T op . It is also sho wn in [S immons, 1982, Lemma 3.7] that, for X exp onentiable, X is stable if and only if, for 12 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN ev ery ultraﬁlter x ∈ U X , the set of all limit p oints of x is ir r educible 1 . F or a nice in tro d u ction to these kinds of s p aces we r efer to [Jun g, 2004]. If we start with a metric compact Hausdorﬀ sp ace X = ( X , d, α ) in stead, the construction ab o v e pro du ces, for ev ery x ∈ U X and x ∈ X , the value of c onver genc e (2. ii ) a ( x , x ) = d ( α ( x ) , x ) ∈ [0 , ∞ ] , whic h brings u s into the realm of 2.5. Approac h spaces. W e will here giv e a quic k ov erv iew of appr o ach sp ac es w h ic h w ere in tro d uced in [Lo we n, 1989] and are extensiv ely describ ed in [Lo wen, 1997]. An approac h space is t ypically deﬁn ed as a pair ( X, δ ) consisting of a set X and an appr o ach distanc e δ on X , that is, a fu nction δ : X × 2 X → [0 , ∞ ] satisfying (1) δ ( x, { x } ) = 0, (2) δ ( x, ∅ ) = ∞ , (3) δ ( x, A ∪ B ) = min { δ ( x, A ) , δ ( x, B ) } , (4) δ ( x, A ) 6 δ ( x, A ( ε ) ) + ε , wh er e A ( ε ) = { x ∈ X | δ ( x, A ) 6 ε } , for all A, B ⊆ X , x ∈ X and ε ∈ [0 , ∞ ]. F or δ : X × 2 X → [0 , ∞ ] and δ ′ : Y × 2 Y → [0 , ∞ ], a map f : X → Y is called appr o ach map f : ( X , δ ) → ( Y , δ ′ ) if δ ( x, A ) > δ ′ ( f ( x ) , f ( A )), for ev ery A ⊆ X and x ∈ X . Approac h spaces and approac h maps are the ob jects and morp hisms of the categ ory A pp . Th e canonical forgetful functor App → Set is top ological, hence App is complete and co complete and App → Set pr eserves b oth limits and colimits. F urtherm ore, the functor App → S et f actors through T op → Set where ( − ) p : App → T op sends a n approac h sp ace ( X , δ ) to the top ological space with th e same underlying s et X and with x ∈ A whenev er δ ( x, A ) = 0 . This fun ctor has a left adjoin t T op → App which one obtains by interpreting th e closure op erator of a top ological space X as δ ( x, A ) = ( 0 if x ∈ A , ∞ else. In fact, the image of this functor can b e describ ed as pr ecisely those approac h spaces where δ ( x, A ) ∈ { 0 , ∞} , for all x ∈ X and A ⊆ X . B eing left adjoin t, T op → A pp preserv es all colimits, and it is not hard to see that this f unctor preserv es also all limits (and hence h as a left adjoint) . As in the case of top olog ical spaces, approac h spaces can b e describ ed in terms of many other concepts suc h as “closed sets” or con verge n ce. F or instance, ev ery appr oac h d istance δ : X × 2 X → [0 , ∞ ] d eﬁnes a map a : U X × X → [0 , ∞ ] , a ( x , x ) = sup A ∈ x δ ( x, A ) , and vice v ersa, eve ry a : U X × X → [0 , ∞ ] deﬁnes a f u nction δ : X × 2 X → [0 , ∞ ] , δ ( x, A ) = inf A ∈ x a ( x , x ) . F urtherm ore, a mapping f : X → Y b et ween approac h sp aces X = ( X , a ) and Y = ( Y , b ) is an app roac h map if and only if a ( x , x ) > b ( U f ( x ) , f ( x )), for all x ∈ U X and x ∈ X . Therefore one migh t take as w ell con vergence as primitiv e notion, and axioms charac terising those fu nctions a : U X × X → [0 , ∞ ] 1 Actually , Lemma 3.7 of [Simmons, 1982] states only one implication, but the other is obvious and even true without assuming exp onen tiability . APPRO ACHING METRIC DOMAINS 13 coming from a appr oac h distance can b e already found in [Low en , 1989]. In this pap er w e will mak e use the c haracterisation (give n in [Clemen tino and Hofmann, 20 03]) as precisely the functions a : U X × X → [0 , ∞ ] s atisfying 0 > a (  x, x ) and U a ( X , x ) + a ( x , x ) > a ( m X ( X ) , x ) , (2. iii ) where X ∈ U U X , x ∈ U X , x ∈ X and U a ( X , x ) = sup A∈ X ,A ∈ x inf a ∈A ,x ∈ A a ( a , x ) . In the language of con vergence, the underlying top ological space X p of an appr oac h space X = ( X, a ) is deﬁned b y x → x ⇐ ⇒ a ( x , x ) = 0, and a top ological sp ace X can b e interpreted as an approac h space b y putting a ( x , x ) = 0 whenev er x → x and a ( x , x ) = ∞ otherwise. W e can restrict a : U X × X → [0 , ∞ ] to principal ultraﬁlters and obtain a m etric a 0 : X × X → [0 , ∞ ] , ( x, y ) 7→ a (  x, y ) on X . Certainly , an approac h m ap is also a metric map, therefore this constr u ction deﬁnes a fu nctor ( − ) 0 : App → Met . whic h , com bined with ( − ) p : Met → Ord , yields a functor App → Ord where x ≤ y whenev er 0 > a (  x, y ). T h is order relation extends p oint-wise to ap p roac h maps, and w e can consider App as an ordered category . As b efore, this additional structure allo ws u s to sp eak ab ou t adjunction in App : for appr oac h m aps f : ( X, a ) → ( X ′ , a ′ ) and g : ( X ′ , a ′ ) → ( X , a ), f ⊣ g if 1 X ≤ g · f and f · g ≤ 1 X ′ ; equiv alentl y , f ⊣ g if and only if, for all x ∈ U X and x ′ ∈ X ′ , a ′ ( U f ( x ) , x ′ ) = a ( x , g ( x ′ )) . One calls an approac h space X = ( X , a ) sep ar ate d , or T 0 , if the underlying top olog y of X is T 0 , or, equiv alentl y , if the u nderlying metric of X is separated. Note that this is the case pr ecisely if, f or all x, y ∈ X , a (  x, y ) = 0 = a (  y , x ) imp lies x = y . Similarly to the situation for metric spaces, b esides the categ orical p r o duct there is a further approac h structure on the set X × Y for approac h sp aces X = ( X , a ) and Y = ( Y , b ), n amely c ( w , ( x, y )) = a ( x , x ) + b ( y , y ) where w ∈ U ( X × Y ), ( x, y ) ∈ X × Y and x = U π 1 ( w ) and y = U π 2 ( w ). The resu lting appr oac h sp ace ( X × Y , c ) w e denote as X ⊕ Y , in fact, one obtains a f unctor ⊕ : App × App → App . W e also n ote that 1 ⊕ X ≃ X ≃ X ⊕ 1, for ev ery app roac h sp ace X . Unfortunately , the ab o ve d escrib ed monoidal structure on App is not closed, the f unctor X ⊕ − : A pp → App do es n ot h a ve in general a r igh t adjoint (see [Hofmann, 2007]). If it do es, we say that the approac h space X = ( X, a ) is + -e xp onentiable and denote this right adjoin t as ( − ) X : App → App . Then, for any approac h space Y = ( Y , b ), the space Y X can b e c h osen as the set of all approac h maps of t yp e X → Y , equipp ed with the con ve rgence (2. iv ) J p , h K = sup { b ( U ev ( w ) , h ( x )) ⊖ a ( x , x ) | x ∈ X , w ∈ U ( Y X ⊕ X ) with w 7→ p , ( w 7→ x ) } , 14 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN for all p ∈ U ( Y X ) and h ∈ Y X . I f p =  k f or some k ∈ Y X , then J  k , h K = sup x ∈ X b 0 ( k ( x ) , h ( x )) , whic h tells u s that ( Y X ) 0 is a subspace of Y X 0 . If X = ( X, a ) happ ens to b e top ologi cal, i.e. a only tak es v alues in { 0 , ∞} , then (2. iv ) sim p liﬁes to J p , h K = sup { b ( U ev ( w ) , h ( x )) | x ∈ X , w ∈ U ( Y X ⊕ X ) with w 7→ p , ( w 7→ x ) , a ( x , x ) = 0 } . F urtherm ore, a top olog ical app r oac h sp ace is +-exp onentiable if and only if it is exp onenti ab le in T op , that is, core-compact. T his follo w s for ins tance from the c h aracterisatio n of exp onentia b le top olog ical spaces giv en in [Pisani, 1999], together with the c haracterisation of +-exp onenti able approac h spaces [Hofmann, 2007] as p recisely th e ones wh ere the conv er gence structur e a : U X × X → [0 , ∞ ] s atisﬁes a ( m X ( X ) , x ) = inf x ∈ X ( U a ( X , x ) + a ( x , x )) , for all X ∈ U U X and x ∈ X . Note that the left hand side is alw a ys smaller or equal to the right hand side. Via the em b edding T op → App describ ed earlier in this sub s ection, whic h is le ft a d join t to ( − ) p : App → T op , w e can in terp ret ev ery top ological space X as an approac h space, also denoted as X , where the con v ergence stru cture tak es only v alues in { 0 , ∞} . Then, for an y approac h sp ace Y , X ⊕ Y = X × Y , whic h in particular tells u s that the diagram App X ⊕− / / App T op O O X ×− / / T op O O comm utes. Therefore, if X is core-compact , th en also th e diagram of the corresp onding right adjoin ts comm utes, hence Lemma 2.3. F or every c or e-c omp act top olo gic al sp ac e X a nd ev e ry appr o ach sp ac e Y , ( Y X ) p = ( Y p ) X . R emark 2.4 . T o b e rigorous, the argument presen ted ab ov e only allo ws us to conclude ( Y X ) p ≃ ( Y p ) X . Ho we ver, since w e can choose the right adjoin ts ( − ) X and ( − ) p exactly as describ ed earlier, one h as indeed equalit y . The lac k of go o d f unction spaces can b e o v ercome by mo ving in to a larger category where these constructions can b e c arried out. In the p articular case of approac h spaces, a goo d en vir on m en t for doing so is the category PsApp of pseudo-appr o ach sp ac es and appr oac h maps [Lo wen and Lo wen, 1989]. Here a pseudo-approac h space is pair X = ( X , a ) consisting of a set X and a c on verge n ce structure a : U X × X → [0 , ∞ ] wh ich only needs to satisfy the ﬁrs t inequalit y of (2. iii ): 0 > a (  x, x ), for all x ∈ X . If X = ( X , a ) and Y = ( Y , b ) are pseudo-appr oac h spaces, then one d eﬁnes X ⊕ Y exactly as for approac h spaces, and th e formula (2. iv ) d eﬁnes a pseud o-approac h structur e on the set Y X of all approac h maps from X to Y , without any fur ther assumptions on X or Y . I n fact, this construction leads now to an adjunction X ⊕ − ⊣ ( − ) X : PsApp → PsApp , for ev ery pseud o-approac h space X = ( X , a ). APPRO ACHING METRIC DOMAINS 15 2.6. Stably compact approac h spaces. Returning to metric compact Hausdorﬀ spaces, one easily v eriﬁes that (2. ii ) d eﬁnes an appr oac h str ucture on X (see [ Tholen, 2009], for instance). Since a ho- momorphism b et w een metric compact Hausdorﬀ sp aces b ecomes an approac h map with resp ect to the corresp ondin g appr oac h structures, one obtains a functor K : MetCompHaus → App . The und erlying metric of K X is just the metric d of the metric compact Hausdorﬀ space X = ( X, d, α ), and x = α ( x ) is a generic c onver genc e p oint of x in K X in the sense that a ( x , y ) = d ( x, y ) , for all y ∈ X . The p oint x is uniqu e up to equiv alence since, if one has x ′ ∈ X with the same pr op ert y , then d ( x, x ′ ) = a ( x , x ′ ) = d ( x ′ , x ′ ) = 0 and, similarly , d ( x ′ , x ) = 0. In analogy to the top ologi cal case, we int r o duce the dual Y op of an approac h space Y = K ( X, d, α ) as Y op = K ( X , d ◦ , α ), and we call an T 0 approac h sp ace stably c omp act if it is of the form K X , for some metric compact Hausdorﬀ space X . Lemma 2.5. L et ( X, d, α ) , ( Y , d ′ , β ) b e metric c omp act Hausdorﬀ sp ac es with c orr esp onding app r o ach sp ac es ( X, a ) and ( Y , b ) , and let f : X → Y b e a map. Then f is an appr o ach map f : ( X, a ) → ( Y , b ) if and only if f : ( X , d ) → ( Y , d ′ ) i s a metric map and β · U f ( x ) ≤ f · α ( x ) , for al l x ∈ U X . Pr o of. Assume ﬁ rst that f : ( X, d ) → ( Y , d ′ ) is in Met and that β · U f ( x ) ≤ f · α ( x ), for all x ∈ U X . Then a ( x , x ) = d ( α ( x ) , x ) > d ′ ( f · α ( x ) , f ( x )) > d ′ ( β · U f ( x ) , f ( x )) = b ( U f ( x ) , f ( x )) . Supp ose no w that f : ( X, a ) → ( Y , b ) is in A pp and let x ∈ U X . T hen 0 > d ( α ( x ) , α ( x )) = a ( x , α ( x )) > b ( U f ( x ) , f · α ( x )) = d ′ ( β · U f ( x ) , f · α ( x )) . Clearly , f : ( X , a ) → ( Y , b ) in App implies f : ( X , d ) → ( Y , d ′ ) in M et , and the assertion follo w s .  It is an imp ortan t fact that K has a left adj oin t M : A pp → MetCompHaus whic h can b e describ ed as follo ws (see [Hofmann, 2010]). F or an app roac h space X = ( X, a ), M X is the metric compact Hausdorﬀ space with underlyin g set U X equipp ed with th e compact Hausdorﬀ con ve r gence m X : U U X → U X and the metric (2. v ) d : U X × U X → [0 , ∞ ] , ( x , y ) 7→ in f { ε ∈ [0 , ∞ ] | ∀ A ∈ x . A ( ε ) ∈ y } , and M f := U f : U X → U Y is a homomorphism b et wee n metric compact Hausdorﬀ sp aces pro vided that f : X → Y is an approac h map b et ween approac h sp aces. The unit and the counit of this adjunction are giv en b y e X : ( X, a ) → ( U X , d ( m X ( − ) , − )) and α : ( U X, d, m X ) → ( X , d, α ) resp ectiv ely , for ( X , a ) in App and ( X, d, α ) in MetCompHaus . R emark 2.6 . All what w as said h ere ab out metric compact Hausd orﬀ spaces and approac h space can b e rep eated, mutatis mutandis , for ordered compact Hausdorﬀ sp aces and top ological spaces. F or instance, the fun cor K : OrdCompHaus → T op (see Subsection 2.4) has a left adjoin t M : T op → OrdCompHaus whic h sends a top olog ical space X to ( U X , ≤ , m X ), where x ≤ y whenev er ∀ A ∈ x . A ∈ y , 16 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN for all x , y ∈ U X . F ur th ermore, L emma 2.5 reads no w a s follo ws: Let ( X , ≤ , α ), ( Y , ≤ , β ) b e ord ered compact Hausdorﬀ spaces with corresp onding top ologica l sp aces ( X , a ) and ( Y , b ), and let f : X → Y b e a map. Then f is a contin u ous map f : ( X, a ) → ( Y , b ) in T op if and only if f : ( X, d ) → ( Y , d ′ ) is in Ord and β · U f ( x ) ≤ f · α ( x ), for all x ∈ U X . Example 2.7. Th e ordered set 2 = { 0 , 1 } with the discrete (compact Hausdorﬀ ) top ology b ecomes an ordered compact Hausdorﬀ space which ind uces th e S ierpi ´ nski sp ace 2 wh ere { 1 } is closed and { 0 } is op en. T hen the maps (1) V : 2 I → 2 (2) v ⇒ − : 2 → 2 , (3) v ∧ − : 2 → 2 are con tinuous, for every set I and v ∈ 2 . F urther m ore (see [Nac hbin, 1992; Escard´ o, 2004]), (4) W : 2 I → 2 is con tinuous if and only if I is a compact top ological space. Here the fu nction space 2 I is p ossibly calculated in th e category PsT op of pseudotop ological spaces (see [Herrlic h et al. , 1991]). In particular, if I is a compact Hausdorﬀ space, then I is exp onen tiable in T op and W : 2 I → 2 b elongs to T op . Example 2.8. T he m etric space [0 , ∞ ] with distance µ ( x, y ) = y ⊖ x equipp ed with the Euclidean compact Hausdorﬀ top ology where x con verge s to ξ ( x ) := sup A ∈ x inf A is a metric compact Hausdorﬀ space (see Example 2.1) whic h giv es the “Sierpi ´ nski approac h space” [0 , ∞ ] with approac h conv er gence structure λ ( x , x ) = x ⊖ ξ ( x ). Then, with the help of subsection 1.1, one sees that (1) sup : [0 , ∞ ] I → [0 , ∞ ], (2) − ⊖ v : [0 , ∞ ] → [0 , ∞ ], (3) − + v : [0 , ∞ ] → [0 , ∞ ] are app r oac h maps, f or ev ery set I and v ∈ [0 , ∞ ]. If I carries the structure a : U I × I → [0 , ∞ ] of an approac h sp ace, one d eﬁnes the de gr e e of c omp actness [Low en, 1997] of I as comp( I ) = sup x ∈ U I inf x ∈ X a ( x , x ) . Then (see [Hofmann, 2007]), (4) inf : [0 , ∞ ] I → [0 , ∞ ] is an appr oac h map if and only if comp( I ) = 0. As ab o ve, the fun ction space [0 , ∞ ] I is p ossibly calculated in PsApp , in fact, ( − ) I : PsApp → PsApp is the righ t adjoint of I ⊕ − : PsApp → PsApp . As any adjun ction, M ⊣ K induces a monad on A pp (resp ectiv ely on T op ). Here, for an y approac h space X , the space K M ( X ) is the set U X of all u ltraﬁlters on the set X equipp ed with an app orac h structure, and the unit and the multiplic ation are essen tially the ones of the ultraﬁlter monad. Therefore w e den ote this monad also as U = ( U, e, m ). In particular, one obtains a functor U := K M : App → A pp (resp ectiv ely U := K M : T op → T op ). S urprisin gly or not, the categories of algebras are equiv alen t to the Eilen b erg–Moore categories on Ord and Met : Ord U ≃ T op U and Met U ≃ App U . More in detail (see [Hofmann, 2010]), for an y metric compact Hausd orﬀ space ( X , d, α ) with corresp ondin g approac h space ( X, a ), α : U ( X, a ) → ( X , a ) is an app r oac h contract ion; an d for an approac h space ( X , a ) with Eilen b erg–Moore algebra structure α : U ( X, a ) → ( X, a ), ( X , d, α ) is a metric compact Hausdorﬀ space where d is the und erlying metric of a and , moreo ve r , a is th e appr oac h structure ind uced by d and α . APPRO ACHING METRIC DOMAINS 17 It is w orthwhile to n ote that th e mon ad U on T op as well as on App satisﬁes a pleasan t tec hnical prop erty: it is of Ko ck- Z¨ ob erlein t yp e [Ko ck, 1995; Z¨ ob erlein, 1976]. In wh at follo ws we will not explore this further an d refer instead f or the deﬁnition and other information to [Es card ´ o and Flagg, 1999 ]. W e just remark here that on e imp ortant consequence of this prop erty is that an Eilen b erg–Moore algebra structure α : U X → X on an { approac h, top ological } space X is necessarily left adjoin t to e X : X → U X . If X is T 0 , then one even has that an approac h m ap α : U X → X is an Eilen b erg–Moore algebra structur e on X if and only if α · e X = 1 X . Hence, a T 0 approac h space X = ( X , a ) is an U -algebra if and only if (1) e very u ltraﬁ lter x ∈ U X has a generic con ve r gence p oint α ( x ) m eaning that a ( x , x ) = a 0 ( α ( x ) , x ), for all x ∈ X , and (2) t he map α : U X → X is an approac h map. W e observed already in [Hofmann, 2010] that th e latter condition can b e sub stituted by (2’) X is +-exp onent iable. F or the reader familiar with the n otion of sob er approac h space [Banasc hewsk i et al. , 2006] we r emark that the former condition can b e sp litted into the follo wing t wo cond itions: (1a) for ev ery ultraﬁlter x ∈ U X , a ( x , − ) is an approac h p rime element, and (1b) X is s ob er. Certainly , the t wo conditions ab ov e im p ly (1). F or the reve r se implication, jus t note that ev ery app roac h prime element ϕ : X → [0 , ∞ ] is the limit f unction of some ultraﬁlter x ∈ U X (see [Banasc hewski et al. , 2006, Prop osition 5.7]). Hence, ev ery stably compact approac h space is sob er. W e call an +-exp onen tiable approac h sp ace X stable if X satisﬁes the condition (1a) ab o v e (compare with Subsection 2.4), an d with this nomenclature one has Prop osition 2.9. A n T 0 appr o ach sp ac e X is stably c omp act if and only if X is sob er, + -exp onentiable and stable. 3. Inject ive app ro ach s p aces 3.1. Y oneda em b eddings. Let X = ( X , a ) b e an appr oac h space with con verge n ce a : U X × X → [0 , ∞ ]. Then a is actually an approac h map a : ( U X ) op ⊕ X → [0 , ∞ ], and we refer to its +-exp onen tial mate y X := p a q : X → [0 , ∞ ] ( U X ) op as the (c ovariant) Y one da emb e dding of X (see [Clemen tino and Hofmann, 2009a] and [Hofmann, 2010]). W e d enote the appr oac h sp ace [0 , ∞ ] ( U X ) op as P X , and its approac h con ve r gence s tructure as J − , − K . One has a ( x , x ) = J U y X ( x ) , y X ( x ) K for all x ∈ U X and x ∈ X (hence y X is indeed an em b ed d ing when X is T 0 ) thanks to the Y one da L emma whic h states h ere that, f or all x ∈ U X and ψ ∈ P X , J U y X ( x ) , ψ K = ψ ( x ) . The metric d : U X × U X → [0 , ∞ ] (see (2. v )) is actually an approac h map d : ( U X ) op ⊕ U X → [0 , ∞ ], whose mate can b e s een as a “se c ond” (c ovariant) Y one da emb e dding Y X : U X → P X , and the “seco n d” Y oneda Lemma reads as (see [Hofmann, 2010]) J U Y X ( X ) , ψ K = ψ ( m X ( X )) , for all X ∈ U U X and ψ ∈ P X . R emark 3.1 . Similarly , the con vergence relation → : ( U X ) op × X → 2 of a top olog ical space X is contin- uous, and by taking its exp onen tial transp ose we obtain the Y oneda em b edd ing y X : X → 2 ( U X ) op . A con tinuous map ψ : X op → 2 can b e identiﬁed with a closed subset A ⊆ U X . In [Hofmann and Tholen, 18 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN 2010] it is sho wn that A corresp onds to a ﬁlter on the lattice of op ens of X , moreo ver, the space 2 ( U X ) op is homeomorphic to th e sp ace F 0 X of all suc h ﬁlters, w here th e top ology on F 0 X h as { f ∈ F 0 X | A ∈ f } ( A ⊆ X op en) as basic op en sets (see [Escard´ o, 1997]). Under this identiﬁca tion, the Y oneda em b edding y X : X → 2 ( U X ) op corresp onds to th e map X → F 0 X sending ev ery x ∈ X to its neighb ou r ho o d ﬁlter, and Y X : U X → F 0 X restricts an ultraﬁlter x ∈ U X to its op en elements. Since an approac h space X is in general not +-exp onenti able, the set [0 , ∞ ] X of all approac h maps of t yp e X → [0 , ∞ ] do es not admit a canonical approac h s tructure. Ho wev er, it still b ecomes a metric space when equipp ed with the sup-metric, that is, the metric space [0 , ∞ ] X is a subspace of the + -exp onen tial [0 , ∞ ] X 0 in Met of underlyin g metric sp ace X 0 of X . Recall from Subsection 1.2 that the con trav ariant Y oneda embed ding h X 0 : X 0 →  [0 , ∞ ] X 0  op of the metric space X 0 sends an elemen t x ∈ X 0 to the metric map X 0 → [0 , ∞ ] , x ′ 7→ a 0 ( x, x ′ ) = a (  x, x ′ ). But the map h X 0 ( x ) can b e also seen as an approac h map of typ e X → [0 , ∞ ], h en ce this construction d eﬁnes also a metric m ap h X : X 0 →  [0 , ∞ ] X  op , for ev ery approac h space X . The inclusion map [0 , ∞ ] X ֒ → [0 , ∞ ] X 0 has a left adjoint [0 , ∞ ] X 0 → [0 , ∞ ] X whic h sends a metric map ϕ : X 0 → [0 , ∞ ] to th e app roac h map X → [0 , ∞ ] whic h send s x to inf x ∈ U X a ( x , x ) + ξ ( U ϕ ( x )) (wh ere ξ ( u ) = sup A ∈ u inf u ∈ A u ). I n particular, if ϕ = a ( e X ( x ) , − ), then ξ ( U ϕ ( x )) = U a ( e U X · e X ( x ) , x ) and therefore inf x ∈ U X a ( x , x ) + ξ ( U ϕ ( x )) = a ( e X ( x ) , − ). Hence, b oth the left and the righ t adjoin t comm ute with the con tra v ariant Y oned a emb eddings.  [0 , ∞ ] X 0  op - -  [0 , ∞ ] X  op m m X 0 h X 0 e e K K K K K K K K K K h X 9 9 t t t t t t t t t t Finally , one also ha ve the Isb el l c onjug ation adjunction in this con text:  [0 , ∞ ] X  op ( − ) − - - ⊤  [0 , ∞ ] X op  0 ( − ) + m m X 0 h X e e J J J J J J J J J J y X 9 9 s s s s s s s s s s where ϕ − ( x ) = s u p x ∈ X ( a ( x , x ) ⊖ ϕ ( x )) and ψ + ( x ) = sup x ∈ U X ( a ( x , x ) ⊖ ψ ( x )) . R emark 3.2 . In our consid er ations ab ov e w e w ere v ery sparse on details and pro ofs. T his is b ecause (in our opinion) this material is b est presented in the language of mo dules (also called distributors or profu nctors), but w e decided not to include this conce pt here and refer for details to [Clemen tino and Hofmann , 2009a] and [Hofmann, 2011] (for the particular con text of this pap er) and to [B ´ enab ou, 2000] and [La wve re , 1973]) for the general concept. W e note that ψ : ( U X ) op → [0 , ∞ ] is the same thing as a m o dule ψ : X − ⇀ ◦ 1 from X to 1 and ϕ : X → [0 , ∞ ] is the same thing as a mo dule ϕ : 1 − ⇀ ◦ X . Then ψ + is the extension of ψ along the iden tity m o dule on X (see [Hofmann, 2011, 1.3 and Remark 1.5]), and ϕ − is the lifting o f ϕ along th e iden tit y m o dule on X (see [Hofmann and W aszkiewicz, 2011, Lemma 5.11]); and this pro cess deﬁnes quite generally an adjunction. APPRO ACHING METRIC DOMAINS 19 3.2. Co complete approac h spaces. I n this and the next s u bsection w e study the notion of c o c omplete- ness for approac h spaces, as initiated in [Clementi no and Hofmann, 2009a; Hofmann, 2011; Clementino and Hofmann, 2009b]. By analogy with ordered sets and metric spaces, we th ink of an approac h map ψ : ( U X ) op → [0 , ∞ ] as a “do wn-set” of X . A p oin t x 0 ∈ X is a supr emum of ψ if a (  x 0 , x ) = sup x ∈ U X a ( x , x ) ⊖ ψ ( x ) , for all x ∈ X . As b efore, sup rema are uniqu e u p to equ iv alence, and therefore w e will often talk ab out the supremum. An app roac h map f : ( X, a ) → ( Y , b ) preserv es the su premum of ψ if b (  f ( x 0 ) , y ) = sup y ∈ U Y b ( U f ( x ) , y ) ⊖ ψ ( x ) . Not surpr isingly (see [Hofmann, 2011]), Lemma 3.3. L eft adjoint appr o ach maps f : X → Y b etwe en appr o ach sp ac es pr eserve al l supr ema which exist in X . W e call an app roac h space X c o c omplete if ev ery “down-set” ψ : ( U X ) op → [0 , ∞ ] has a supremum in X . If this is the case, then “taking supr ema” deﬁnes a map S up X : P X → X , in deed, one h as Prop osition 3.4. An appr o ach sp ac e X is c o c omplete if and only if y X : X 0 → ( P X ) 0 has a lef t adjoint Sup X : ( P X ) 0 → X 0 in Met . R emark 3.5 . W e deviate here from the notatio n u s ed in p revious w ork wh ere a sp ace X was calle d co complete whenev er y X : X → P X has a left adjoint in App . Approac h sp aces satisfying this (stronger) condition will b e called abs olutely co complete (see S ubsection 3.4 b elo w) h ere. With th e help of Subsection 3.1, one sees immediatel y that Sup X : ( P X ) 0 → X 0 pro du ces a left inv er s e of h X 0 : X 0 →  [0 , ∞ ] X 0  op in Met , hence the underlying metric space X 0 is complete. Certainly , a left in verse of X 0 →  [0 , ∞ ] X 0  op in M et giv es a left in verse of y X : X 0 → ( P X ) 0 in M et , h ow ev er, su ch a left in verse do es n ot need to b e a left adjoin t (see Example 3.9). In the follo wing sub section w e will see what is missing. 3.3. Sp ecial types of colimits. Similarly to what w as done for metric spaces, we will b e intereste d in approac h sp aces w hic h admit certain t yp es of s u prema. Our ﬁrst example are tensor e d appr oac h spaces whic h are deﬁn ed exact ly as their metric coun terparts. Explicitly , to ev ery p oin t x of an approac h space X = ( X, a ) and ev ery u ∈ [0 , ∞ ] one a sso ciates a “do wn-set” ψ : ( U X ) op → [0 , ∞ ] , x 7→ a ( x , x ) + u , and a suprem u m of ψ (whic h, w e recall, is u nique up to equ iv alence) w e denote as x + u . Then X is called tensored if ev ery suc h ψ has a supr em um in X . By deﬁnition, x + u ∈ X is c haracterised by the equation a ( e X ( x + u ) , y ) = sup x ∈ U X ( a ( x , y ) ⊖ ( a ( x , x ) + u )) , for all y ∈ X . Th is suprem u m is actually obtained for x =  x , so that the righ t h and side ab o ve reduces to a (  x, y ) ⊖ u . Therefore: Prop osition 3.6. An appr o ach sp ac e X is tensor e d if and only if its underlying metric sp ac e X 0 is tensor e d. 20 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN W e call an approac h space X = ( X, a ) U -c o c omplete if ev ery “down-set” ψ : ( U X ) op → [0 , ∞ ] of the form ψ = Y X ( x ) with x ∈ U X has a supr em um in X . Su c h a supremum, denoted as α ( x ), is c h aracterised b y a (  α ( x ) , x ) = sup y ∈ U X ( a ( y , x ) ⊖ d ( y , x )) , for all x ∈ X . Here the supr em um is obtained for y = x , hence the equalit y ab ov e translates to a 0 ( α ( x ) , x ) = a ( x , x ) , where a 0 denotes the underlying metric of a . S ince a ( x , x ) = d ( x ,  x ) where d is the metric on ( U X ) 0 , we conclude that Prop osition 3.7. An appr o ach sp ac e X i s U -c o c omplete if and only if e X : X 0 → ( U X ) 0 has a left adjoint α : ( U X ) 0 → X 0 in Met . Note that ev ery metric compact Hausdorﬀ space is U -cocomplete. W e are no w in p ositio n to c h aracterise co complete approac h sp aces. Theorem 3.8. L et X b e an appr o ach sp ac e . Then the fol lowing assertions ar e e quivalent. (i) X i s c o c omplete. (ii) The metric sp ac e X 0 is c omplete and and the appr o ach sp ac e X is U -c o c omplete. (iii) The metric sp ac e X 0 is c omplete and e X : X 0 → ( U X ) 0 has a left adjoint α : ( U X ) 0 → X 0 in Met . F urthermor e, in this situation the supr emum of a “down-set” ψ : ( U X ) op → [0 , ∞ ] i s g iven by (3. i ) _ x ∈ U X α ( x ) + ψ ( x ) . Pr o of. T o see the implication (iii) ⇒ (i), we only need to show that the form ula (3. i ) give s indeed a supremum of ψ . In fact, a 0 ( _ x ∈ U X α ( x ) + ψ ( x ) , x ) = su p x ∈ U X a 0 ( α ( x ) + ψ ( x ) , x ) = s up x ∈ U X ( a 0 ( α ( x ) , x ) ⊖ ψ ( x )) = sup x ∈ U X ( a ( x , x ) ⊖ ψ ( x )) , for all x ∈ X .  Example 3.9. E very metric compact Hausdorﬀ sp ace whose underlyin g metric is co complete giv es r ise to a co complete app roac h space. In particular, b oth [0 , ∞ ] and [0 , ∞ ] op are cocomplete (see Example 2.1). T o eac h m etric d on a set X on e asso ciat es the approac h con v ergence stru cture a d ( x , y ) = sup A ∈ x inf y ∈ A d ( x, y ) , and this constru ction deﬁn es a left adjoin t to the forgetful f unctor ( − ) 0 : App → Met . F urthermore, note that a d (  x, y ) = d ( x, y ). In particular, for the metric space [0 , ∞ ] = ([0 , ∞ ] , µ ) one obtains a µ ( x , y ) = su p A ∈ x inf x ∈ A ( y ⊖ x ) , and the approac h sp ace ([0 , ∞ ] , a µ ) is not U -cocomplete. T o see this, consider an y ultraﬁlter x ∈ U [0 , ∞ ] whic h con tains all sets { x ∈ [0 , ∞ ] | u ≤ x < ∞} , u ∈ [0 , ∞ ] and u < ∞ . Then a µ ( x , ∞ ) = ∞ and a µ ( x , y ) = 0 for all y ∈ [0 , ∞ ] with y < ∞ , hence a µ cannot b e of the form µ ( α ( − ) , − ) f or a map α : U [0 , ∞ ] → [0 , ∞ ]. Ho we ver, for the metric s pace APPRO ACHING METRIC DOMAINS 21 [0 , ∞ ] op = ([0 , ∞ ] , µ ◦ ), the appr oac h con v ergence str u cture a µ ◦ is actually the structure indu ced b y the metric compact Hausdorﬀ space ([0 , ∞ ] , µ ◦ , ξ ) and therefore ([0 , ∞ ] , a µ ◦ ) is co complete. U -cocomplete approac h spaces are closely related to m etric compact Hausd orﬀ spaces resp ectiv ely stably compact approac h space, in b oth cases the app roac h s tructure a on X can b e d ecomp osed into a metric a 0 and a map α : U X → U , and one reco v ers a as a ( x , x ) = a 0 ( α ( x ) , x ). In fact, ev ery metric compact Hausdorﬀ space is U -cocomplete, b ut the rev erse implication is in general false since, for instance, the map α : U X → X do es not need to b e an Eilen b erg–Moore algebra structure on X (i.e. a compact Hausdorﬀ top ology) . F ortunately , this p rop erty of α wa s not needed in the pro of of Lemma 2.5, and w e conclude Lemma 3.10. L et ( X , a ) and ( Y , b ) b e U -c o c omplete appr o ach sp ac es and f : X → Y b e a map. Then f : ( X, a ) → ( Y , b ) is an appr o ach map if and only if f : ( X, a 0 ) → ( Y , b 0 ) is a metric map and, for al l x ∈ U X , β · U f ( x ) ≤ f · α ( x ) . R emark 3.11 . Once again, ev ery th ing told here has its topological coun terpart. F or instance, w e call a top ological s pace X U -cocomplete whenev er the monotone map e X : X 0 → ( U X ) 0 has a left adj oint α : ( U X ) 0 → X 0 in Ord . Th en, with ≤ denoting the underlying order of X , an ultraﬁlter x ∈ U X con ve r ges to x ∈ X if and only if α ( x ) ≤ x . Moreo ver, one also h as an analog ve r sion of th e lemma abov e. Recall from Su b section 2.5 that ( − ) p : App → T op denotes the canonical forgetful fu nctor from A pp to T op , where x → x in X p if and only if 0 = a ( x , x ) in the approac h space X = ( X , a ). If X = ( X , a ) is also U -cocomplete w ith left adjoin t α : ( U X ) 0 → X 0 , then, f or any x ∈ U X and x ∈ X , α ( x ) ≤ x ⇐ ⇒ 0 = a 0 ( α ( x ) , x ) ⇐ ⇒ 0 = a ( x , x ) ⇐ ⇒ x → x. Here ≤ denotes the un derlying order of the underlying top ology of X , whic h is the same as the un d erlying order of the underlying metric of X . Hence, α pro vid es also a left adjoin t to e X : X p 0 → U ( X p ) 0 , and therefore the top ological space X p is U -cocomplete as wel l. An imp ortant consequence of this f act is Prop osition 3.12. L et X = ( X, a ) and Y = ( Y , b ) b e U -c o c omplete app r o ach sp ac es and f : X → Y b e a map. Then f : ( X , a ) → ( Y , b ) is an appr o ach map if and only if f : ( X, a 0 ) → ( Y , b 0 ) is a metric map and f : X p → Y p is c ontinuous. Finally , w e also obs erv e th at U -cocomplete approac h spaces are stable u nder standard constructions: b oth X ⊕ Y and X × Y are U -cocomplete, provi ded that X = ( X , a ) and Y = ( Y , b ) are so. 3.4. Op-con tinuous lattices w it h an [0 , ∞ ] -action. W e call an appr oac h space X absol u tely c o c omplete if the Y oneda em b edding y X : X → P X has a left adjoin t in App . This is to sa y , X is co complete and the metric left adjoin t Sup X of y X is actually an appr oac h map Sup X : P X → X . It is sho wn in [Hofmann , 2011] that • the absolutely co complete approac h spaces are precisely the in jectiv e ones, and that • the category InjApp sup of absolutely co complete approac h T 0 spaces and su premum pr eserving (= left adjoin t) approac h maps is m onadic o v er App , M et and S et . The construction X 7→ P X is the ob ject part of the left adjoin t P : A pp → InjApp sup of the inclusion fun ctor InjApp sup → App , and the maps y X : X → P X deﬁne the u n it y of the induced monad P = ( P , y , m ) on App . C omp osing this monad with the adjunction ( − ) d ⊣ ( − ) : A pp ⇆ Set giv es th e corresp onding monad on Set . 22 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN This resembles very m uc h w ell-kno wn prop erties of injectiv e top ologi cal T 0 spaces, w h ic h are kno wn to b e the algebras for the ﬁ lter monad on T op , Ord and S et , hence, b y Remark 3.1, are precisely the (accordingly deﬁned) ab s olutely co complete top olog ical T 0 spaces. F u rthermore, all in formation ab out the top ology of an in jectiv e T 0 space is conta in in its underlying ord er , and the ordered sets o ccurring this wa y are th e op-c ontinuous lattic es , i.e. the duals of con tinuous lattices 2 , as sh o wn in S cott [1972] (see S ubsection 2.1). In the sequel we will write ContLat ∗ to denote th e category of op-co ntin uous lattices and maps preserving all suprema and do wn-dir ected inﬁm a. Note that ContLat ∗ is equ iv alen t to the category of absolutely co complete top ologi cal T 0 spaces and left adjoin ts in T op , and of course also to the category ContLat of con tinuous lattices and m ap s pr eserving u p-directed s uprema and all inﬁm a. These an alogies make us conﬁdent that abs olutely co complete approac h T 0 spaces pr o vide an in teresting metric coun terpart to (op-)con tinuous lattices. In fact, in [Hofmann, 2010] it is s h o wn that the approac h structure of su c h a sp ace is determined by its un d erlying metric, hen ce we are talking essentiall y ab out metric sp aces h ere. Moreo ver, ev ery absolutely co complete approac h space is exp onen tiable in A pp and the ful l sub cate gory of A pp deﬁned by these spaces is Cartesian closed. Th eorem 3.20 b elo w exp oses no w a tigh t connection w ith op-contin u ous lattice s : the absolutely co complete approac h T 0 spaces are p recisely the op-con tinuous lattice s equipp ed with an unitary and a s so ciativ e action of [0 , ∞ ] in the monoidal catego r y ContLat ∗ . Ev ery app roac h space X = ( X, a ) indu ces appr oac h maps X ⊕ [0 , ∞ ] B X − − → P X , U X Y X − − → P X , X I F X,I − − − → P X (where I is compact Hausdorﬀ ) . Exactly as in Subsection 1.3, B X : X ⊕ [0 , ∞ ] → P X is the mate of the comp osite ( U X ) op ⊕ X ⊕ [0 , ∞ ] a ⊕ 1 − − − → [0 , ∞ ] ⊕ [0 , ∞ ] + − → [0 , ∞ ] , and F X,I : X I → P X is the mate of the comp osite ( U X ) op ⊕ X I → [0 , ∞ ] I inf − − → [0 , ∞ ] , where the ﬁrst comp onent is the mate of the comp osite ( U X ) op ⊕ X I ⊕ I 1 ⊕ ev − − − → ( U X ) op ⊕ X a − → [0 , ∞ ] . Explicitely , for ϕ ∈ X I and x ∈ U X , F X,I ( ϕ )( x ) = inf i ∈ I a ( x , ϕ ( i )). A s u premum of the “down-set” F X,I ( ϕ ) ∈ P X is n ecessarily a supremum of the family ( ϕ ( i )) i ∈ I in the underlying order of X . If X is co complete, one can comp ose the maps ab o v e with Sup X and obtains metric maps X 0 ⊕ [0 , ∞ ] + − → X 0 , ( U X ) 0 α − → X 0 , ( X I ) 0 W − → X 0 ( I compact Hausdorﬀ ) , (3. ii ) whic h are eve n morphism s in App pro vid ed th at X is absolutely co complete. In fact, one has Prop osition 3.13. L et X b e an appr o ach sp ac e. Then X is absolutely c o c omplete if and only if X is c o c omplete and the thr e e maps ( 3. ii ) ar e appr o ach map s. Pr o of. The obtain the rev ers e imp lication, w e ha ve to sh o w that the mapping Sup X : P X → X , ψ 7→ _ x ∈ U X ( α ( x ) + ψ ( x )) is an approac h map. W e w r ite X d for the discrete app r oac h space with un derlying set X , then U ( X d ) is just a compact Hausd orﬀ space, namely the ˇ Cec h-Stone compactiﬁcation of the set X . By assu mption, W : X U ( X d ) → X is an approac h map, th er efore it is enough to show that U ( X d ) ⊕ P X → X, ( x , ψ ) 7→ α ( x ) + ψ ( x ) 2 Recall th at our und erlying order is d ual to the sp ecialisation order. APPRO ACHING METRIC DOMAINS 23 b elonges to A pp . Sin ce th e diagonal ∆ : U ( X d ) → U ( X d ) ⊕ U ( X d ) as well as the identit y maps U ( X d ) → U X and U ( X d ) → ( U X ) op are in A pp , we can express the map ab o ve as the comp osite U ( X d ) ⊕ P X ∆ ⊕ 1 − − − → U X ⊕ ( U X ) op ⊕ P X α ⊕ ev − − − − → X ⊕ [0 , ∞ ] + − → X of approac h m aps.  Example 3.14. The approac h space [0 , ∞ ] is inj ectiv e and hence absolutely co complete, bu t [0 , ∞ ] op is not injectiv e. T o see this, either observe that the map f : { 0 , ∞} → [0 , ∞ ] op , 0 7→ ∞ , ∞ 7→ 0 cannot b e extended along the subspace in clus ion { 0 , ∞} ֒ → [0 , ∞ ], or that the mappin g ( u, v ) 7→ u ⊖ v (whic h is the tens or of th e metric space [0 , ∞ ] op ) is not an approac h map of type [0 , ∞ ] op ⊕ [0 , ∞ ] → [0 , ∞ ] op . T herefore [0 , ∞ ] op is n ot absolutely co complete, ho wev er, recall from Examp le 3.9 that [0 , ∞ ] op is co complete. R emark 3.15 . S imilarly , a top ological sp ace X is absolutely co complete if and only if X is co complete and the latter t wo maps of (3. ii ) (accordingly deﬁned) are con tinuous. Lemma 3.16. L et X b e an appr o ach sp ac e and I b e a c omp act Hausdorﬀ sp ac e. If X is c o c omplete and X p is absolutely c o c omplete, then X I is U -c o c omplete. Pr o of. W e w r ite a : U X × X → [0 , ∞ ] for the con vergence structure of the approac h s p ace X , and b : U I × I → 2 for the conv er gence structure of the compact Hausd orﬀ space I . In th e b oth cases there are maps α : U X → X and β : U I → I resp ectiv ely so that a ( x , x ) = a 0 ( α ( x ) , x ) and b ( u , i ) = true if and only if β ( u ) = i , for all x ∈ U X , x ∈ X , u ∈ U I and i ∈ I . F or every p ∈ U ( X I ) and h ∈ X I , J p , h K = sup { a 0 ( α ( U ev ( w )) , h ( β ( u )) | w ∈ U ( X I × I ) , w 7→ p , ( w 7→ u ) } = sup i ∈ I sup w ∈ U ( X I × I ) U π 1 ( w )= p β · U π 2 ( w )= i a 0 ( α ( U ev ( w )) , h ( i )) = sup i ∈ I a 0 ( _ w ∈ U ( X I × I ) U π 1 ( w )= p β · U π 2 ( w )= i α ( U ev ( w )) , h ( i )) = sup i ∈ I a 0 ( γ ( p )( i ) , h ( i )) , where w e deﬁ n e γ ( p )( i ) = _ w ∈ U ( X I × I ) U π 1 ( w )= p β · U π 2 ( w )= i α ( U ev ( w )) . In order to conclude that γ is a map of t yp e U ( X I ) → X I , we h a ve to sho w that γ ( p ) is a con tin uous map γ ( p ) : I → X p , for ev ery p ∈ U ( X I ). T o this end , w e note ﬁrst that the sup rem u m ab o ve can b e rewritten as γ ( p )( i ) = _ w ∈ U ( X I × I ) U π 1 ( w )= p α ( U ev( w )) & b ( U π 2 ( w ) , i ) . W e put Y = { w ∈ U ( X I × I ) | U π 1 ( w ) = p } and consider Y as a subspace of U (( X I × I ) d ), that is, the ˇ Cec h-Stone compactiﬁcatio n of the set X I × I . No te that Y is compact, and one has con tinuous maps Y U ev − − − → U ( X d ) , Y U π 2 − − − → U ( I d ) , U ( X d ) α − → X p , U ( I d ) × I b − → 2 . 24 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN Therefore we can exp ress the map γ ( p ) as the comp osite I − → X I p W − → X p of con tinuous maps , wh ere the ﬁ rst comp onent is the mate of the comp osite Y × I ∆ × 1 − − − → Y × Y × I U ev × U π 2 × 1 − − − − − − − − → U ( X d ) × U ( I d ) × I α × b − − − → X p × 2 & − → X p of con tinuous maps .  Prop osition 3.17. L et X = ( X, d ) b e a c o c omplete metric sp ac e whose underlying or der e d set X p is a op-c ontinuous lattic e. Then ( X, d, α ) is a metric c omp act Hausdorﬀ sp ac e wher e α : U X → X is deﬁne d by x 7→ ^ A ∈ x _ x ∈ X x . Pr o of. Since X p is op-con tin u ous, X p is ev en an ordered compact Hausdorﬀ space with con verge n ce α . W e hav e to sho w that α : U ( X, d ) → ( X , d ) is a metric map. Recall from Lemma 2.2 that with ( X, d ) also U ( X, d ) is tensored, h ence we can apply Prop osition 1.10. Firstly , f or x ∈ U X and u ∈ [0 , ∞ ], α ( x ) + u =   ^ A ∈ x _ x ∈ X x   + u ≤ ^ A ∈ x _ x ∈ X ( x + u ) = α ( x + u ) since − + u : X → X p reserv es suprema. S econdly , let x , y ∈ U X and assume 0 = U d ( x , y ) = sup A ∈ x ,B ∈ y inf x ∈ A,y ∈ B d ( x, y ) = su p B ∈ y inf A ∈ x sup x ∈ A inf y ∈ B d ( x, y ) . F or the last equalit y see [Seal, 2005, Lemma 6.2] , for instance. W e wish to sh o w that α ( x ) ≤ α ( y ), that is, ^ A ∈ x _ x ∈ A x ≤ ^ B ∈ y _ y ∈ B y , whic h is equiv alen t to ^ A ∈ x _ x ∈ A x ≤ _ y ∈ B y , for all B ∈ y . L et B ∈ y and ε > 0. By h yp othesis, th er e exist some A ∈ x with su p x ∈ A inf y ∈ B d ( x, y ) < ε , h ence, for all x ∈ A , there exist some y ∈ B with d ( x, y ) < ε and therefore x + ε ≤ y . C onsequen tly , for all ε > 0,   ^ A ∈ x _ x ∈ A x   + ε ≤ ^ A ∈ x _ x ∈ A ( x + ε ) ≤ _ y ∈ B y ; and therefore also ^ A ∈ x _ x ∈ A x ≤ _ y ∈ B y .  Theorem 3.18. L et X b e a T 0 appr o ach sp ac e. Then the f ol lowing assertions ar e e qui v alent. (i) X i s absolutely c o c omplete. (ii) X is c o c omplete, X p is absolutely c o c omplete and + : X ⊕ [0 , ∞ ] → X is an appr o ach map. (iii) X is c o c omplete, X p is absolutely c o c omplete and + : X p × [0 , ∞ ] p → X p is c ontinuous. (iv) X is U -c o c omplete, X 0 is c o c omplete, X p is absolutely c o c omplete, a nd, for al l x ∈ X and u ∈ [0 , ∞ ] , the map − + u : X → X pr eserves down-dir e cte d inﬁma and the map x + − : [0 , ∞ ] → X sends up-dir e cte d supr ema to down-dir e cte d inﬁma. Pr o of. Clearly , (i) ⇒ (ii) ⇒ (iii) ⇒ (i v ). Assume now (iv). According to Prop osition 3.13, w e ha ve to sho w that the thr ee maps (3. ii ) are approac h maps. W e write a : U X × X → [0 , ∞ ] for th e conv ergence structure of the appr oac h sp ace X , by h yp othesis, a ( x , x ) = d ( α ( x ) , x ) where d is the u n derlying metric and α ( x ) = ^ A ∈ x _ x ∈ X x . By Lemma 3.17, ( X, d, α ) is a metric compact Hausdorﬀ space and therefore APPRO ACHING METRIC DOMAINS 25 α : U X → X is an appr oac h map. Since the metric space X 0 is cocomplete, + : X 0 ⊕ [0 , ∞ ] → X 0 and W : X I 0 → X 0 ( I any s et) are metric maps. If I is a compact Hausdorﬀ space, ( X I ) 0 is a subsp ace of X I d 0 , therefore also W : ( X I ) 0 → X 0 is a metric map. F u rthermore, sin ce X p is abs olutely co complete, W : ( X I ) p = ( X p ) I → X p is contin uous (see Lemma 2.3 ). Since X I is U -cocomplete by Lemma 3.16, W : X I → X is actually an approac h map by Pr op osition 3.12. S imilarly , + : ( X ⊕ [0 , ∞ ]) 0 → X 0 is a m etric map since ( X ⊕ [0 , ∞ ]) 0 = X 0 ⊕ [0 , ∞ ]. O ur h yp othesis states that + : ( X ⊕ [0 , ∞ ]) p = X p × [0 , ∞ ] p → X p is contin u ous in eac h v ariable, and [Scott, 1972, Prop osition 2.6] tells us th at it is indeed conti nuous. By applying Prop ositio n 3.12 again we conclude that + : X ⊕ [0 , ∞ ] → X is an app roac h map.  Note that the approac h structure of an absolutely co complete T 0 approac h sp ace can b e reco ve r ed from its u n derlying m etric since th e con verge nce α : U X → X is d eﬁned b y the und erlying lattice structure. In fact, th e Theorem ab o v e sho w s that an a b solutely cocomplete T 0 approac h sp ace is e s sen tially the same thin g as a separated co complete metric space X = ( X , d ) wh ose und er lyin g ordered set is an op-con tin uous lattice (see Prop osition 3.17) and wh ere the action + : X × [0 , ∞ ] → X preserv es do wn- directed inﬁma (in b oth v ariables). In the ﬁn al part of this pap er w e com bin e this w ith Theorem 1.8 where separated co complete m etric s paces are describ ed as s up-lattices X equipp ed with an u nitary an d asso ciativ e action + : X × [0 , ∞ ] → X on the set X which preserv es su p rema in eac h v ariable, or, equiv alentl y , + : X ⊗ [0 , ∞ ] → X is in S up . F or X , Y , Z in ContLat ∗ , a map h : X × Y → Z is a b imorphism if it is a morp h ism of ContLat ∗ in eac h v ariable. Prop osition 3.19. The c ate gory ContLat ∗ admits a tensor pr o duct which r epr esents bimorphism s. That is, for al l X , Y in ContLat ∗ , the functor Bimorph( X × Y , − ) : ContLat ∗ → Set is r epr esentable by some obje ct X ⊗ Y in ContLat ∗ . Pr o of. One easily v eriﬁes that Bimorph( X × Y , − ) preserves limits. W e c h ec k the solution set cond ition of F reyd’s Adjoin t F u nctor Th eorem (in the f orm of [MacLane, 1971, Section V.3, Theorem 3]). T ake S as an y representi ng set of { L ∈ ContLat ∗ | | L | ≤ | F ( X × Y ) |} , where F ( X × Y ) denotes the set of all ﬁlters on the s et X × Y . Let Z b e an op-con tin u ous lattice and ϕ : X × Y → Z be a b imorphism. W e ha ve to ﬁnd some L ∈ S , a b imorphism ϕ ′ : X × Y → L and a morp hism m : L → Z in ContLat ∗ with m · ϕ ′ = ϕ . Since the map e : X × Y → F ( X × Y ) sending ( x, y ) to its p rincipal ﬁlter giv es actually the reﬂection of X × Y to ContLat ∗ , there exists some f : F ( X × Y ) → Z in ContLat ∗ with f · e = ϕ . F ( X × Y ) q $ $ $ $ I I I I I f   L z z m z z t t t t t t X × Y e A A                  ϕ / / ϕ ′ = q · e 4 4 Z Let f = m · q a (regular epi,mono)-factorisation of f in ContLat ∗ . T hen ϕ ′ := q · e is a bimorphism as it is the corestriction of ϕ to L , m : L → Z lies in ContLat ∗ and L can b e chosen in S .  By un icit y of the representi ng ob ject, 1 ⊗ X ≃ X ≃ X ⊗ 1 and ( X ⊗ Y ) ⊗ Z ≃ X ⊗ ( Y ⊗ Z ). F ur thermore, with th e order > , [0 , ∞ ] is actually a mon oid in ContLat ∗ since + : [0 , ∞ ] × [0 , ∞ ] → [0 , ∞ ] is a bimorphism 26 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN and therefore it is a m orp hism + : [0 , ∞ ] ⊗ [0 , ∞ ] → [0 , ∞ ] in ContLat ∗ , and so is 1 → [0 , ∞ ] , ⋆ 7→ 0. W e write ContLat ∗ [0 , ∞ ] for the category whose ob jects are op-conti n u ous lattices X equipp ed with a unitary and asso ciativ e act ion + : X ⊗ [0 , ∞ ] → X in ContLat ∗ , and wh ose morphisms are those ContLat ∗ -morphisms f : X → Y whic h satisfy f ( x + u ) = f ( x ) + u , for all x ∈ X and u ∈ [0 , ∞ ]. Summing up, Theorem 3.20. InjApp sup is e quivalent to ContLat ∗ [0 , ∞ ] . Here an absolutely cocomplete T 0 approac h sp ace X = ( X, a ) is sen t to its underlying ordered set where x ≤ y ⇐ ⇒ a (  x, y ) = 0 ( x, y ∈ X ) equ ip p ed with the tensor pr o duct of X , and an op-cont in u ous lattice X with action + is sent to the approac h space induced by th e metric compact Hausd orﬀ space ( X, d, α ) where d ( x, y ) = in f { u ∈ [0 , ∞ ] | x + u ≤ y } and α ( x ) = ^ A ∈ x _ x ∈ A x , for all x, y ∈ X and x ∈ U X . R emark 3.21 . By the theorem ab o ve, the d iagram InjApp sup ≃ ContLat ∗ [0 , ∞ ] ⊣   ⊥ / / ContLat ∗ ⋌ w w o o o o o o o o o o o o o o o o o o o o o o o o o o o −⊗ [0 , ∞ ] t t Set P D D F A A of righ t adjoin ts commutes, and therefore the diagram of the (dotted) left adjoin ts do es so to o. Here F X is the set of all ﬁ lters on the s et X , ordered by ⊇ , and P X = [0 , ∞ ] U X where U X is equipp ed with the Zariski top ology . In other words, P X ≃ F X ⊗ [0 , ∞ ], for ev ery set X . Referen ces Banaschewski, B. , Lowen, R. a nd V an Olmen, C. (20 06), Sob er approach spaces, T op olo gy Appl. 1 53 (16), 3059– 3070 . B ´ enabou, J. (2000), Distributors at w or k, lecture notes b y Thomas Streic her ( http:/ /www.m athem atik.tu- darmstadt.de/ ~ streic her/ ). Bonsangue, M. M. , v an Breugel, F. and R utten, J. J. M. M. (1998 ), Generalized metric spa ces: completion, top ology , and pow er domains via the Yoneda embedding, The or et. Comput. Sci. 193 (1-2), 1–51 . Clementino, M. M. and Hofmann, D. (200 3), T op ologica l features o f lax algebras, Appl. Cate g. St ructur es 11 (3), 267–286 . Clementino, M. M. and Hofmann, D. (200 9a), Lawvere completeness in Topo logy , Appl. Cate g. Struct u r es 17 , 175–2 10, ar Xiv:ma th.CT /0704.3976 . Clementino, M. M. and Hofmann, D. (2009 b), Relative injectivity as co co mpleteness fo r a cla ss of distributors , The ory Appl. Cate g. 21 , No. 12, 210–23 0 , a rXiv:m ath.C T/0807.4123 . Da y, A . (1975), Filter monads, co ntin uous la ttices and closure systems, Canad . J . Math. 27 , 50–59 . Eilenberg, S. and Kell y, G. M. (196 6), Closed c a tegories , in Pr o c. Conf. Cate goric al Algebr a (L a J ol la, Calif., 1965) , pages 421–56 2 , Springer, New Y ork . APPRO ACHING METRIC DOMAINS 27 Escard ´ o, M. H. (1997 ), Injective spaces via the ﬁlter monad, in Pr o c e e dings of the 12th Sum m er Confer enc e on Gener al T op olo gy and its Applic ations (North Bay, O N , 1997) , volume 22 , pages 97 –100. Escard ´ o, M. H. (2004), Sy nthetic top olo gy of da ta types and cla ssical spaces, Ele ctr onic Notes in The or et ic al Computer Scienc e 87 , 21–15 6. Escard ´ o, M. H . and Flagg, R. (19 99), Semantic do mains, injective spaces and monads., Bro okes, Stephen (ed.) et al., Mathematical foundations o f progra mming seman tics . Pr o ceedings of the 15th c o nference, T ula ne Univ., New O rleans, LA, April 28 - May 1, 1999 . Amster dam: Elsevier , Electronic Notes in Theor etical Computer Science. 20, electronic pap e r No.15 (19 99). Flagg, R. C. (1992), Completeness in contin uity spaces., Seely , R. A. G. (ed.), Categor y theory 1 991. Pro ceeding s of an international summer categ ory theory meeting, held in Mon tr´ ea l, Qu´ eb e c, Canada, June 2 3-30, 1991. Providence, RI: American Mathematical So ciety . CMS Conf. Pro c. 13, 183 -199 (1992). Flagg, R. C. (1997a), Algebraic theories of compa ct p o spaces, T op olo gy Appl. 77 (3), 277–29 0. Flagg, R. C. (1997b), Quant ales and contin uity spaces, Algebr a Universalis 37 (3), 257–276 . Flagg, R. C. a nd Koppe rma n, R. (1997 ), Contin uity spaces: r econciling domains a nd metric spaces, The or et. Comput. Sci. 177 (1), 11 1–13 8, ma thematical foundations of programming seman tics (Manhatta n, K S, 1994). Flagg, R. C. , S ¨ underhauf, P. and W a gner, K. (1996), A logica l appro ach to quantitativ e domain theory , T op olo gy Atlas Pr eprint # 23 http://at. yorku .ca/e/a/p/p/23.htm . Gierz, G. , Hofmann, K. H. , Keimel, K. , La wson, J. D. , Mislove, M. W. and Scott, D. S. (20 0 3), Continuous lattic es and domains , v o lume 93 o f Encyclop e dia of Mathematics and its Applic ations , Ca mbridge Univ ersity Pr ess, C a mbridge, xxxvi+5 91 pages. Herrlich, H. , Lowen-C olebunders, E. and Schw arz, F. (1 991), Improving To p: PrTop and PsTop, in Cate gory the ory at work (Br emen, 1990) , volume 18 of R es. Exp. Math. , pages 21–34, Heldermann, Berlin. Hofmann, D. (200 7), T op olo g ical theories and close d o b jects, A dv. Math. 215 (2), 789–824. Hofmann, D. (201 0), Duality for distributiv e spaces, T e chnical repo rt, arXiv:m ath.CT /1009.3892 . Hofmann, D. (201 1 ), Injectiv e s paces via a djunction, J. Pur e Appl. Algebr a 215 (3), 283– 302, arXiv: math. CT/08 04.0326 . Hofmann, D. and Tholen, W. (20 10), Lawvere completion and s eparation via closure, Appl. Cate g. S tructur es 18 (3), 259–287 , arXiv :math .CT/0801.0199 . Hofmann, D . and W aszkiewicz, P. (20 11), Approximation in quantale-enriched categories, ac c epte d for public a- tion in T op olo gy Appl., arXiv: math.C T/1004.2228 . Johnstone, P. T. (1986), Stone sp ac es , volume 3 of Cambridge S tudies in A dvanc e d Mathematics , Ca m bridge Univ ersity Pr ess, C a mbridge, xxii+37 0 pa g es, reprint o f the 1982 edition. Jung, A . (2004), Stably compact s paces and the probabilistic p ow er space construction, in J . Desharnais and P . Panangaden, editors, Domain-the or etic Metho ds in Pr ob abilistic Pr o c esses , volume 8 7, 15pp. Kell y, G. M. (1982 ), Basic c onc epts of enrich e d c ate gory the ory , v olume 64 o f L ondon Mathematic al So ciety L e ctur e Note S eries , Cambridge Universit y Press, Cambridge, 245 pa ges, also in: Repr. Theor y Appl. Categ. No. 10 (2005), 1–136. Kell y, G. M. a nd Schmitt, V. (2005), Notes on enriched c a tegories with colimits of some class, The ory A ppl. Cate g. 14 , No. 17, 3 99–4 23. Kock, A. (1995 ), Mo nads fo r which structures are adjoint to units, J. Pur e Appl. Algebr a 104 (1), 41–59 . Kopp erman , R. (1988), All topolo g ies come from ge ne r alized metrics, A mer. Math. Monthly 95 (2), 89–97 . 28 GONC ¸ ALO GUTIERRES AN D DIRK H OFMANN Kost anek, M. and W aszkiewicz, P. (2011), The formal ball mo del for Q -categ ories, Math. Struct ur es Comput. Sci. 21 (1), 41–64. La wvere, F. W. (1973), Metr ic spaces, g eneralized logic, a nd clos ed ca tegories, R end. Sem. Mat. Fis. Milano 43 , 135–1 66 (19 7 4), a lso in: Repr. Theory Appl. Categ. No. 1 (200 2), 1–37. Lowen, E. and Lowen, R. (1989 ), T op olog ical quasitop os hulls of categor ies containing top o logical and metric ob jects, Cahiers T op olo gie G´ eom. Diﬀ´ er en t iel le Cat´ eg. 30 (3 ), 21 3–22 8. Lowen, R. (1989), Approach spa ces: a common superc a tegory of T O P and ME T, Math. Nachr. 141 , 183–2 2 6. Lowen, R. (1997), Appr o ach sp ac es , Oxford Mathematica l Mo nogra phs, The Clar endon Press Oxford Univ e r - sity Pr ess, New Y o rk, x+ 253 pages, the missing link in the top ology -uniformity-metric tr iad, Oxford Science Publications. MacLane, S. (1971), Cate gories for the working mathematician , Springer-V erla g, New Y ork, ix+26 2 pa ges, grad- uate T ex ts in Ma thematics, V o l. 5. Manes, E. G. (1969), A t riple theoretic construc tio n of compact alg ebras., Sem. on T riples a nd Categorical Homology Theory , ETH Z ¨ uric h 196 6 /67, Le c t. No tes Ma th. 80, 9 1-118 (1969). Nachbin, L. (195 0), T op olo gia e Ord em , Univ. of Chicago Press, in Portug ue s e, English translation: T o p o lo gy a nd Order, V a n Nostrand, Princeton (1965). Nachbin, L. (199 2), Compact unions of closed subsets are c losed and co mpact intersections of op en subsets are op en, Portugal. Math. 49 (4), 403– 409. Pedicchio, M. C. and Tholen, W. (198 9 ), Multiplicative structures ov er s up- lattices, Ar ch. Math. (Brno) 25 (1- 2), 107–1 14. Pisani, C. (1999), Conv er gence in exp onentiable spa ces, The ory Appl. Cate g. 5 , No. 6, 148–1 62. R utten, J. J. M. M. (1998 ), W eighted co limits a nd for mal balls in gener a lized metric spaces, T op olo gy A ppl. 89 (1-2), 179–2 02, domain theory . Scott, D. (1972 ), Co ntin uous lattices, in T op oses, algebr aic ge ometry and lo gic (Conf., Dalhousie U n iv., Halifax, N. S., 1971) , pages 97–136 . Lecture Notes in Math., V ol. 274, Springer, Berlin. Seal, G. J. (20 05), Canonical and op- c anonical lax a lgebras , The ory Appl. Cate g. 14 , 221 – 243. Simmons, H. (198 2), A couple of triples, T op olo gy Appl. 13 (2), 201– 223. Tholen, W. (2009 ), Or de r ed topolo gical structures, T op olo gy Appl. 156 (12), 2148–2 1 57. Vickers, S. (2005 ), Lo ca lic completion of ge neralized metric spaces. I, The ory Appl. Cate g. 14 , No. 15, 3 28–35 6. W a gner, K. R. (1994), Solving R e cursive Domain Equations with Enriche d Cate gories , Ph.D. thesis , Carnegie Mellon Universit y , ftp:// ftp.ri sc.uni- linz.ac.at/pub/techreports/1994/94- 62.ps.gz . W aszkiewicz, P. (2009), On domain theo ry ov er Girard qua nt a les, F u nd. Inform. 92 (1-2), 169–19 2. Wyler, O. (1985 ), Alg e braic theories for co nt inuous se milattices, Ar ch. Ra t ional Me ch. Anal . 90 (2), 9 9–113 . Z ¨ oberlein, V. (1976 ), Do ctrines on 2-categ ories, Math. Z. 148 (3), 267–27 9. CMUC, Dep ar tme nt of Ma thema tics, University of Coimbra, 3001-454 Coimbra, Por tuga l E-mail addr ess : ggutc@mat.uc.pt CIDMA, Dep ar tment of Ma thema tics, Universi ty of A vei ro , 3810-193 A veiro, Por tuga l E-mail addr ess : dirk@ua.pt

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