Injective Spaces via Adjunction

INJECTIVE SP A CE S VIA ADJUNCTION DIRK HOFMANN A bstra ct . Our work ov er the past years sho ws that not only the collection of (for instance) all topological spaces gi ves rise to a cate gory , but also ea ch topological space can be seen indi vidually as a catego ry by interpreting the co n ver gence relation x − → x be tween ultrafilters and points of a to pological space X as arro ws in X . Naturally , this point of vie w opens the door to the use of concepts and i deas from (enriched) Category Theory for the inv estigation of (for instance) topological spaces. In this paper we study cocomp leteness, adjoint functors and Kan extension s in the con text of topolo gical theories. W e sho w that the cocomplete spaces are precisely the injective spaces, and the y are algebras f or a suitable monad on Set . This way we obtain enriched versions of kno wn results about injective topological spaces and continuous lattices. I ntr oduction The title of the present article is clearly remin iscent of the cha pter O r der ed sets via adjunction s by R. W ood [W oo0 4], where the theory of ordered sets is de velo ped ele gantl y employing conseq uently the concept of adjunction. One of the funda mental aspects of our rec ent resear ch is des cribed by the slogan topolo gical spac es ar e cate gories , and therefore can be studied using notions and techniqu es from (enri ched) C ateg ory Theory . W e hope to be able to sho w in this paper that con cepts like module, colimit and adjoi ntness can be a very useful tool for the study of topolo gical spa ces too. W e sho uld explain what is meant by “spaces are categorie s”. In his famou s 1973 paper [Law73] F .W . Lawve re c onsider s the poin ts of a (g eneralis ed) m etric space X as the ob jects of a cate gory X and l ets the distan ce d ( x , y ) ∈ [0 , ∞ ] play the role of the hom-se t of x and y . In fact, the basic laws 0 ≥ d ( x , x ) and d ( x , y ) + d ( y , z ) ≥ d ( x , z ) remind us immediat ely to the operations “choosing the identity” and “compositio n” 1 − → hom( x , x ) and hom( x , y ) × hom( y , z ) − → hom ( x , z ) of a category . Moti v ated by Lawv ere’ s approach, we consid er the points of a topologic al space X as the object s of our cate gory , and in terprete the con ver gence x − → x of an ultr afilter x on X to a point x ∈ X as a morphis m in X . Wit h this interpreta tion, the con ver gen ce relation ( ∗ ) − → : U X × X − → 2 becomes the “hom-functo r” of X . Clearly , we hav e to make here the concess ion tha t a morphism in X does no t ha v e just an ob ject b ut rathe r an ultrafilte r (of objec ts) as domain. This intuit ion is suppor ted by 2000 Mathematics Subject Classification. 18A05, 18D1 5, 18D20, 18B35, 18C15, 54B30, 54A20. K e y wor ds and phrases. Quantale, V -category , monad, topological theory , module, Y oneda lemma, weighted colimit. The author ackno w ledges partial financial assistance by Unidade de Inv estigac ¸ ˜ ao e Desen v olvimento Matem ´ atica e Aplicac ¸ ˜ oes da Univ ersidade de A veiro / FCT . 1 2 DIRK HOFMANN the obs erv ation (due to M. Barr [Bar70]) that a relation x − → x be tween ultrafilters and points of a set X is the con ver genc e relatio n of a (un ique) topology on X if and only if e X ( x ) − → x and ( X − → x & x − → x ) | = m X ( X ) − → x , ( † ) for all x ∈ X , x ∈ U X an d X ∈ U U X , where m X ( X ) is the fi ltered sum of the filters in X and e X ( x ) =  x the principal ultrafilter generated by x ∈ X . In the second axiom we use the natural extensio n of a relatio n between ultrafilters and points to a relation between ultrafilters of ultrafilter s and ultrafilters, so that X − → x is a meanin gful expressio n. In our int erpretati on, the first condit ion postul ates the exi stence of an “identity arrow” on X , w hereby the second one requires the ex istence of a “compo site” of “composable pairs of arro ws”. Furthermor e, a function f : X − → Y between topological spaces is contin uous whene ver x − → x in X implies f ( x ) − → f ( x ) in Y , that is, f associates to each object in X an object in Y and to each arrow in X an arrow in Y betwee n the correspondi ng (ultrafilter of) objects in Y . It is now a lit tle step to admit that the hom-fun ctor ( ∗ ) of such a cate gory X takes v alues in a quantal e V other than the two-eleme nt Boolean algebra 2 , and that the domain x of an arro w x − → x in X is an element of a set T X other than the set U X of all ultrafilters of X . A s one can see immediately , we need T to be a functo r T : Set − → Set in ord er to define the notion of func tor between such cate gories, moreo ver , we need T to be part of a Set -monad T = ( T , e , m ) in order to formu late the axioms ( † ) of a cate gory in th is conte xt. Eventu ally , we re ach the no tion of a ( T , V )-cate gory (also c alled ( T , V )-algeb ra or lax alge bras), for a Set -monad T and q uantale V , as int roduced in [CH03, CT03, CHT 04]. A di ff erent b ut related approach to this kind of categ ories was presente d by Burroni [Bur71]. Though the initial paper [CH 03] focused on the topological featur es of this approa ch, already in [CT03] the emphasi s was put on the categ orical descriptio n of ( T , V )-algeb ras. T he theory of cate gories enrich ed in a monoidal close d categ ory V is by no w classical [Ben63, Ben6 5, EK66, Kel82, Law73]. W e hav e a wide range of concepts and theorems at our disposal, it includes such things as modules (also called distri bu tors, profuncto rs), weighted (co )limits, the Y oneda L emma, K an extensi ons, adjoint functo rs, and man y more. Naturally , we wish to lift these notio ns and resul ts to the ( T , V )-settin g. A first step in this direction was done in [CH 07], where the notion of module is introduc ed int o th e realm of (no w called) ( T , V )-cate gories . A s in the case o f V -cat egor ies, this c oncept i s fu ndamental f or t he fur ther de vel opment of the theory ; for instance, completenes s proper ties of ( T , V )-cate gories are formulated in terms of modul es. In fac t, in [CH07] the categ orical notion of Cauch y-comple teness (the na me Lawv ere- complete ness respec ti vely L-complete ness is proposed in [C H07, HT08]) is introduce d and studied. A furthe r achie vemen t of [CH07] is the formulat ion and proof of a ( T , V )-ve rsion of the famous Y oneda lemma, a result which turns out to be crucial for the study of ( T , V )-cate gories in the same way as the classic al result is for the dev elopmen t of the theory of V -categor ies. This can be judged by looking at the result s an d pr oofs of the su bsequen t paper [HT08] and also the present one. Howe v er , in order to procee d with our “spaces as categorie s” pro ject, furth er conditions on the monad T and the quantale V are needed . As a result of our work on this subject emerged the notion of a topolog ical theory T = ( T , V , ξ ) introd uced in [Hof07 ], where we add a map ξ : T V − → V compat ible with the monad and the quanta le structu re to our set ting. Our exper ience sho w s so far that this co ncept is broad enou gh to include our princi pal e xamples, an d at the same time restri cti ve enou gh to allo w us to introduce cat egor ical ideas into the real m of ( T , V )-cate gories (which we now cal l T -cat egor ies). The particula r topic of this paper is the stud y of weighted colimits, cocomplete T -cat egori es and adjoin t T -fun ctors. W e start by recalling the definit ion of the pri ncipal players, namely T -categorie s, T -functors and T -mo dules, and then proceed introduci ng adjoin t T -functors and weighted colimits for T -catego ries precisely as for V -categ ories. Furthermore , we show that the de velo pment of many basic INJECTIVE SP ACES VIA ADJUNCTION 3 proper ties does not go much be yond the V -cate gory case, as soo n as we ha ve T -substitutes for dual cate- gory , preshe af-const ruction and t he Y oneda l emma a v ailable . F inding usefu l equi v alents to the se not ions and results w e see as one of the main challenges here, fortunate ly , most of these problems are already solv ed in [CH07]. Howe ve r , in thi s paper we giv e a di ff erent approach to the Y oneda lemma, by pro ving a more general result (Theorem 1.10 ) more suitable for our purpose. Moreo ver , our pro of does not need any more the restricti ve condition T 1 = 1. The achie v ements of this paper can then be summarised as follo ws. W e characterise coco mplete T -categ ories as precisel y the injecti ve ones with res pect to ful ly fait hful T -functors , and as those T -cate gories X for w hich the Y oneda functor y X : X − → ˆ X into the preshe af T -c atego ry ˆ X has a left adjoint . W e deduc e cocompleten ess of the presheaf T -category ˆ X , and sho w the exi stence of K an-e xtensio ns in our setting, that is, any T -functor f : X − → Y into a cocom- plete T -category has an (up to equi v alence) unique extensio n to a left adjoint T -fu nctor f L : ˆ X − → Y . As a con sequenc e, we see that the category T - Cocont sep of sep arated and cocomple te ( = injecti ve ) T - cate gories and le ft adjoint T -functo rs is a reflect iv e subc atego ry of T - Cat (and o f T - Cat sep ), the category of (separate d) T -c atego ries and T - functors . F urther more, we sho w tha t th e in duced m onad on T - Cat sep is of Kock- Z ¨ oberle in type and the inclus ion functor is e ven monadic. W e also prov e that t he for getful func- tors fro m T - C ocont sep to S et and to V - Cat sep are monad ic. A t this point we notice that our cate goric al approa ch has led us t o a w ell-kno wn res ult for topolog ical spac es: injecti ve T 0 -space s (t ogether with suit- able m orphis ms) are the E ilenber g–Moore algebras for the “filter on open subse ts” monad on T op 0 , the cate gory of T 0 -space s and cont inuous maps, as well as for the fi lter m onad on Set (see [Day75, E sc97] for details ). W e hav e now generalised thes e fa cts to T -categ ories, b ut to do so we used (almost) only standa rd argumen ts from Categ ory Theory! Finally , w e wish to highlight a possib le applic ation of our work. O ne of the nice features of do main theory is the strong interacti on between topologic al and order -theoretic ideas. For instance, continuous lattice s [Sco7 2] can be des cribed purely in order theoretic terms as well as in topolog ical terms: as ordere d sets with certain completeness prop erties, or as injecti v e topolog ical T 0 -space s with respect to embeddings . There exis t m any interesting attempts in the literature to introduc e continu ous metric spaces , or , more genera l, continu ous V -cate go ries ; all of them are (more or less) based on the orde r- theore tic appro ach to continuou s lattices ([W ag94, B vBR98, W as02]). W e are not aware of an y attempt using injecti vity properties in a suitable cate gory . T he res ults of our work indicate that, for instanc e, R. Lowen’ s appro ach space s ([Lo w97]) can serv e as a useful tool for the intr oductio n and stud y of contin uous metric spaces. In fact, as a particul ar insta nce of our work we deduc e that the injecti ve T 0 -appro ach spaces can be described as the cocomplete T 0 -appro ach spaces, bu t also as the Eilenber g– Moore alg ebras for suit able monads on sets r especti v ely metric spaces. L ooking at it f rom the ot her end, we o btain a metric equi v alen t to the filter mo nad, whose alg ebras are prec isely the injecti ve T 0 -appro ach spaces . 1. T h e S e tting 1.1. T opological theories. Throughout this paper w e conside r a (stric t) to polo gical the ory as introduced in [Hof07]. Such a theory T = ( T , V , ξ ) con sists of a commuta tiv e qua ntale V = ( V , ⊗ , k ), a Set -monad T = ( T , e , m ) w here T and m satisfy (BC) (tha t is, T sends pullbacks to weak pullback s and each natura lity square of m is a weak pullback) and a m ap ξ : T V − → V such that (1) the monoid V i n Set lifts to a monoid ( V , ξ ) in ( Set T , × , 1), that is, ξ : T V − → V is a T -algeb ra structu re on V and ⊗ : V × V − → V and k : 1 − → V are T -algeb ra homomorp hisms. In orth er 4 DIRK HOFMANN words , w e require the follo wing diagrams to commute. X e X / / 1 X B B B B B B B B T X ξ   X T T X m X   T ξ / / T X ξ   T X ξ / / X 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 (2) ξ X : = ξ · T ( − ) defines a natural transformati on ( ξ X ) X : P V − → P V T : Set − → Ord . Here P V : Set − → Ord is the V -po werset functor defined as foll ows. W e put P V ( X ) = V X with the pointwis e order . For a fun ction f : X − → Y , we ha ve a monot one map V f : V Y − → V X , ϕ 7− → ϕ · f . It is easy to see that V f preser ves all infima and all suprema, hence has in particula r a left adjoint denoted as P V ( f ). Explicitly , for ϕ ∈ V X we ha v e P V ( f )( ϕ )( y ) = W { ϕ ( x ) | x ∈ X , f ( x ) = y } . Examples 1.1. (1) The identi ty theory I = ( 1 , V , 1 V ), for each quantale V , where 1 = (Id , 1 , 1) denote s the id entity monad. (2) U 2 = ( U , 2 , ξ 2 ), where U = ( U , e , m ) denotes the ultrafilter monad and ξ 2 is essential ly the identi ty 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 theor y ( L , V , ξ ⊗ ), for each quanta le 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 1.2. V -rela tions. The quantaloi d V - Rel [BCSW 83] has s ets as objects , and an ar ro w r : X − → 7 Y from X to Y is a V -r elat ion r : X × Y − → V . Compo sition of V -relations r : X − → 7 Y and s : Y − → 7 Z is defined as matrix multipli cation s · r ( x , z ) = _ y ∈ Y r ( x , y ) ⊗ s ( y , z ) , and the identity arro w 1 X : X − → 7 X is the V -relation which sends all diagon al elemen ts ( x , x ) to k and all other elements to the bottom elemen t ⊥ of V . The complete orde r of V induces a complete order on V - Rel ( X , Y ) = V X × Y : for V -relations r , r ′ : X − → 7 Y w e define r ≤ r ′ : ⇐ ⇒ ∀ x ∈ X ∀ y ∈ Y . r ( x , y ) ≤ r ′ ( x , y ) . Any ele ment u ∈ V can be interpre ted as a V -rela tion u : 1 − → 7 1. T hen, gi ve n also v ∈ V , v · u = v ⊗ u , and k represents the identity arro w . W e hav e an in v olut ion ( r : X − → 7 Y ) 7− → ( r ◦ : Y − → 7 X ) w here r ◦ ( y , x ) = r ( x , y ), satisfying 1 ◦ X = 1 X , ( s · r ) ◦ = r ◦ · s ◦ , r ◦ ◦ = r , as well as r ◦ ≤ s ◦ whene ve r r ≤ s . Furthermore , there is an obviou s functor Set − → V - Rel , ( f : X − → Y ) 7− → ( f : X − → 7 Y ) INJECTIVE SP ACES VIA ADJUNCTION 5 sendin g a map f : X − → Y to its graph f : X − → 7 Y defined by f ( x , y ) =          k if f ( x ) = y , ⊥ else. Then, in the quantal oid V - Rel , we hav e f ⊣ f ◦ . If the quantale V is non-tri vial, i.e. if ⊥ < k , then the functo r abov e from Set to V - Rel is faith ful and we can identify the function f : X − → Y with the V -relatio n f : X − → 7 Y . In the sequ el w e will always assume ⊥ < k , and write f : X − → Y for both the functi on and the V -relation. Let t : X − → 7 Z be a V -relation. The compositio n func tions − · t : V - Rel ( Z , Y ) − → V - Rel ( X , Y ) and t · − : V - Rel ( Y , X ) − → V - Rel ( Y , Z ) . preser ve suprema and therefore ha ve respecti v e right adjoints ( − )  t : V - Rel ( X , Y ) − → V - Rel ( Z , Y ) and t  ( − ) : V - Rel ( Y , Z ) − → V - Rel ( Y , X ) . Hence, for V -relatio ns s : Z − → 7 Y , r : X − → 7 Y re specti v ely s : Y − → X , r : Y − → 7 Z , we ha ve bijec tions s · t ≤ r and t · s ≤ r . s ≤ r  t s ≤ t  r X  ? ? ? ? r   ? ? ? ? _ t   Z ≤  s / / Y Z X _ t O O ≤ Y  ? ? ? ? r _ _ ? ? ? ?  s o o W e call r  t the ext ension of r alo ng t , and t  r the lifting of r alon g t . 1.3. T -r elations. The functor T : S et − → Set extends to a 2-f unctor T ξ : V - Rel − → V - Rel as follo ws: we put T ξ X = T X for each set X , and T ξ r : T X × T Y − → V r ( x , y ) 7− → _  ξ · T r ( w )     w ∈ T ( X × Y ) , T π 1 ( w ) = x , T π 2 ( w ) = y  for each V -relation r : X − → 7 Y . T hat is, T ξ r : T X × T Y − → V is the smallest (ord er -preserv ing) map s : T X × T Y − → V such that ξ · T r ≤ s · can. T ( X × Y ) can / / ξ X × Y ( r ) = ξ · T r # # H H H H H H H H H T X × T Y T ξ r { { V ≤ As sho wn in [Hof07], we hav e T ξ f = T f for each function f : X − → Y , T ξ ( r ◦ ) = T ξ ( r ) ◦ (and we write T ξ r ◦ ) for each V -relat ion r : X − → 7 Y , m beco mes a natural transformation m : T ξ T ξ − → T ξ and e an op-lax natura l transformation e : Id − → T ξ , i.e. e Y ◦ r ≤ T ξ r ◦ e X for all r : X − → 7 Y in V - Rel . A V -relation of the form α : T X − → 7 Y we call T -r elati on from X to Y , and w rite α : X − ⇀ 7 Y . For T -relations α : X − ⇀ 7 Y and β : Y − ⇀ 7 Z we define the Kleisli con volution β ◦ α : X − ⇀ 7 Z as β ◦ α = β · T ξ α · m ◦ X . Kleisli con vo lution is associati ve and has the T -r elation e ◦ X : X − ⇀ 7 X as a lax identity: a ◦ e ◦ X = a and e ◦ Y ◦ a ≥ a for any a : X − ⇀ 7 Y . W e call a : X − ⇀ 7 Y unita ry if e ◦ Y ◦ a = a , so that e ◦ X : X − ⇀ 7 X is the identity on X in the catego ry T - URel of sets and unitary T -relatio ns, with the Kleisli con v olution 6 DIRK HOFMANN as composition. In fact, T - URel is a loca lly completely 2-cate gory , where the 2-cat egor ical structu re is inheri ted from V - Rel . Furthermore, for a T -relatio n α : X − ⇀ 7 Y , the compo sition function − ◦ α still has a right adjo int ( − )  α but α ◦ − in gener al not . Explicitly , gi v en also γ : X − ⇀ 7 Z , we pas s from X  γ / _ α  Z Y to T X  γ / / _ m ◦ X   Z T T X _ T ξ α   T Y and define γ  α : = γ  ( T ξ α · m ◦ X ). One easily ve rifies the requ ired univ ersal property , which in particu lar implies that γ  α is unitary if α and γ are so. 1.4. T -ca tegorie s. A T -cate gory is a pair ( X , a ) consisting of a set X and a T -e ndorela tion a : X − ⇀ 7 X on X such that e ◦ X ≤ a and a ◦ a ≤ a . Expresse d elementwise, these condition s become k ≤ a ( e X ( x ) , x ) and T ξ a ( X , x ) ⊗ a ( x , x ) ≤ a ( m X ( X ) , x ) for all X ∈ T T X , x ∈ T X and x ∈ X . A functi on f : X − → Y between T -catego ries ( X , a ) and ( Y , b ) is a T -functor if f · a ≤ b · T f , w hich in pointwise notatio n read s as a ( x , x ) ≤ b ( T f ( x ) , f ( x )) for all x ∈ T X , x ∈ X . If w e hav e abo ve e v en equ ality , we call f : X − → Y ful ly faithfu l . The resulti ng cate gory of T -cat egori es and T -fun ctors we denote as T - C at . T he quant ale V becomes a T -cat egor y V = ( V , hom ξ ), where hom ξ : T V × V − → V , ( v , v ) 7− → hom( ξ ( v ) , v ) (see [Hof07]). Examples 1.2. (1) For each quantale V , I V -cate gories are precisely V -categ ories and I V -funct ors are V -functors. As usual, we w rite V -cate gory instead of I V -cate gory , V -functor instead of I V - functo r , and V - Cat instead of I V - Cat . (2) The main result of [Bar70] states that U 2 - Cat is isomorphic to the categ ory T op of topological spaces and continuo us maps. In [CH03] it is sho wn that U P + - Cat is isomorph ic to the categor y App of app roach spaces and non-exp ansi ve maps [Low97]. The category Set T of T -algeb ras and T -homomorp hisms can be embed ded into T - Cat by regardin g the structure map α : T X − → X of an Eilenber g–Moo re algebra ( X , α ) as a T -relation α : X − ⇀ 7 X . The T -catego ry resu lting this way from the free E ilenber g–Moore algebra ( T X , m X ) we denote as | X | . The for getfu l functor O : T - Cat − → Set , ( X , a ) 7− → X is topo logi cal (see [AHS90]), hence has a left and a right adjoint and T - Cat is complete and cocomp lete. T he free T -cate gory on a set X is giv en by ( X , e ◦ X ). In pa rticular , the free T - categ ory (1 , e ◦ 1 ) on a o ne-elemen t set is a generator in T - Cat w hich we den ote as G = (1 , e ◦ 1 ). W e hav e a canonic al for getful functor S : T - Cat − → V - Cat sendi ng a T -cate gory X = ( X , a ) to its underlying V -catego ry S X = ( X , a · e X ). Furthermore, S has a left adjoint A : V - Cat − → T - Cat defined by A X = ( X , e ◦ X · T ξ r ), for each V -cate gory X = ( X , r ). Ho we ver , there is yet anothe r interesting functo r connectin g T -cate gorie s with V -categ ories, namely M : T - Cat − → V - Cat which sends a T - cate gory ( X , a ) to the V -category ( T X , T ξ a · m ◦ X ). This funct ors are used in [CH07] to define the dual of a T -catego ry X : X op = A (M( X ) op ) . INJECTIVE SP ACES VIA ADJUNCTION 7 Clearly , if T = I V is the identity t heory I V = ( 1 , V , 1 V ), then X op is the u sual dual V -cate gory of X . It is by no means ob vious why the definition a bov e pro vides us w ith a “good ” gen eralisati on of th is const ruction. W e take Theorem 1.9 as well as the Y oneda lemma for T -categorie s (see Theorem 1.10 and C orollar y 1.11) as a reaso n to believ e so. As studied in [Hof07], the te nsor product of V ca n be transport ed to T - Cat by puttin g ( X , a ) ⊗ ( Y , b ) = ( X × Y , c ) with c ( w , ( x , y )) = a ( x , x ) ⊗ b ( y , y ) , where w ∈ T ( X × Y ), x ∈ X , y ∈ Y , x = T π 1 ( w ) a nd y = T π 2 ( w ). The T -categ ory E = (1 , k ) is a ⊗ -neutral object , w here 1 is a singleton set and k : T 1 × 1 − → V the c onstant relatio n with valu e k ∈ V . In genera l, this construc tions does not result in a closed structure on T - Cat ; ho wev er , the results of [Hof07 ] giv e us the foll owin g Pro position 1.3. F or each T -alg ebra X , X ⊗ − : T - Cat − → T - Cat has a right adjoint ( − ) X : T - Cat − → T - Cat . In particul ar , the struct ur e ~ − , −  on V | X | is given by the formula ~ p , ψ  = ^ q ∈ T ( | X |× V | X | ) q 7− → p hom( ξ · T ev( q ) , ψ ( m X · T π 1 ( q ))) , for eac h p ∈ T V | X | and ψ ∈ V | X | . More over , for p = e V | X | ( ϕ ) we have ~ e V | X | ( ϕ ) , ψ  = ^ x ∈ T X hom( ϕ ( x ) , ψ ( x )) . Furthermor e, se v eral maps obtaine d from the qua ntale structure on V become no w T -functors. Pro position 1.4. The following assertion s hold. (1) Both k : E − → V a nd ⊗ : V ⊗ V − → V a r e T -functors , hence V is e ven a monoid in ( T - Cat , ⊗ , E ) . (2) ξ : | V | − → V is a T -functor . (3) W : V | X | − → V is a T -functo r , for eac h set X . Pr oo f. (1) and (2) are easy to pro ve, (3) is a conse quence of [Hof07, Proposition 6.11].  1.5. T -modules. Let X = ( X , a ) and Y = ( Y , b ) be T -cat egori es and ϕ : X − ⇀ 7 Y be a T -re lation. W e call ϕ a T -modu le , and write ϕ : X − ⇀ ◦ Y , if ϕ ◦ a ≤ ϕ and b ◦ ϕ ≤ ϕ . Note that we always hav e ϕ ◦ a ≥ ϕ and b ◦ ϕ ≥ ϕ , so tha t the T -module condit ion abov e implies equali ty . Kleisli con v olution is associ ati ve, and it follo ws that ψ ◦ ϕ is a T -mod ule if ψ : Y − ⇀ ◦ Z and ϕ : X − ⇀ ◦ Y are so. Furthermo re, we hav e a : X − ⇀ ◦ X for each T -ca tego ry X = ( X , a ), and, by definition , a is the identity T -mod ule on X for the Kleisli con vo lution. In other words, T -catego ries and T -modules form a cate gory , denoted as T - Mod , with Kleisli con v olut ion as comp ositiona l structu re. In fact, T - Mod is an orde red cate gory w ith the structu re on hom -sets inherited from T - Rel . A s before , a I V -module we call simply V -module and write ϕ : X − → ◦ Y , and pu t V - Mod = I V - Mod . F inally , a T - relation ϕ : X − ⇀ 7 Y is unita ry pre cisely if ϕ is a T -module ϕ : ( X , e ◦ X ) − ⇀ ◦ ( Y , e ◦ Y ) between the corre spondin g discrete T -cat egor ies. Remark 1.5 . S ince the compo sitional and the order structu re for T -modules is as for T -re lations, for each T -module ϕ : ( X , a ) − ⇀ ◦ ( Y , b ) and each T -ca tego ry Z = ( Z , c ) we hav e an ord er -preserv ing map − ◦ ϕ : T - Mod ( Y , Z ) − → T - Mod ( X , Z ). One easi ly ver ifies that , if ζ : ( X , a ) − ⇀ ◦ ( Z , c ) i s a T - modules, then so is ζ  ϕ . Hence, − ◦ ϕ has a right adjoi nt ( − )  ϕ . Furthermore, if ϕ ⊣ ψ in T - Mod , then − ◦ ψ ⊣ − ◦ ϕ in Ord , and ther efore − ◦ ϕ = ( − )  ψ . 8 DIRK HOFMANN Let now X = ( X , a ) and Y = ( Y , b ) be T -cate gories and f : X − → Y be a function. W e define T - relatio ns f ∗ : X − ⇀ 7 Y and f ∗ : Y − ⇀ 7 X by putting f ∗ = b · T f and f ∗ = f ◦ · b respect iv ely . Hence, for x ∈ T X , y ∈ T Y , x ∈ X and y ∈ Y , we ha ve f ∗ ( x , y ) = b ( T f ( x ) , y ) and f ∗ ( y , x ) = b ( y , f ( x )). G i ven now T -modules ϕ and ψ , we obtain ϕ ◦ f ∗ = ϕ · T f and f ∗ ◦ ψ = f ◦ · ψ. In particul ar , b ◦ f ∗ = f ∗ and f ∗ ◦ b = f ∗ , as well as f ∗ ◦ f ∗ = b · T f · T f ◦ · T ξ b · m ◦ Y ≤ b . The follo w ing lemma can be easil y verified. Lemma 1.6. The following asser tions ar e equivalen t. (i) f : X − → Y is a T -f unctor . (ii) f ∗ is a T -module f ∗ : X − ⇀ ◦ Y . (iii) f ∗ is a T -module f ∗ : Y − ⇀ ◦ X . (i v) a ≤ f ∗ ◦ f ∗ . As a con sequenc e, for eac h T -fu nctor f : ( X , a ) − → ( Y , b ) we ha v e an adju nction f ∗ ⊣ f ∗ in T - Mod . Moreo ver , giv en also a T -functor g : ( Y , b ) − → ( Z , c ), g ∗ ◦ f ∗ = c · T g · T f = c · T ( g · f ) = ( g · f ) ∗ and f ∗ ◦ g ∗ = f ◦ · g ◦ · c = ( g · f ) ◦ · c = ( g · f ) ∗ . Since also (1 X ) ∗ = (1 X ) ∗ = a , w e obt ain functors ( − ) ∗ : T - Cat − → T - Mod and ( − ) ∗ : T - Cat op − → T - Mod , where X ∗ = X = X ∗ , for each T -category X . Lemma 1.7. A T -functor f : ( X , a ) − → ( Y , b ) is fully faithful if and only if 1 ∗ X = f ∗ ◦ f ∗ . Lemma 1.8. Consider T -modules ϕ : X − ⇀ ◦ Y , ψ : X − ⇀ ◦ Z and α : Y − ⇀ ◦ B, wher e α is right adjoint. Then α ◦ ( ϕ  ψ ) = ( α ◦ ϕ )  ψ. Pr oo f. Let β : B − ⇀ ◦ Y be the left adjoint of α . W e hav e to show that the diag ram T - Mod ( X , Y ) ( − )  ψ / / α ◦−   T - Mod ( Z , Y ) α ◦−   T - Mod ( X , B ) ( − )  ψ / / T - Mod ( Z , B ) of right adjo ints commutes. But the diagram T - Mod ( X , Y ) T - Mod ( Z , Y ) −◦ ψ o o T - Mod ( X , B ) β ◦− O O T - Mod ( Z , B ) β ◦− O O −◦ ψ o o of the correspon ding left adjoints co mmutes since Kleisli con v oluti on is ass ociati v e, and the assertion follo ws.  INJECTIVE SP ACES VIA ADJUNCTION 9 Theor em 1.9 ([CH07]) . F or T -cate go ries ( X , a ) and ( Y , b ) , and a T -r elation ψ : X − ⇀ 7 Y , the following assert ions ar e equivalen t. (i) ψ : ( X , a ) − ⇀ ◦ ( Y , b ) is a T -module. (ii) Both ψ : | X | ⊗ Y − → V an d ψ : X op ⊗ Y − → V ar e T -functo rs. Therefore , eac h T -mod ule ϕ : X − ⇀ ◦ Y defines a T -fun ctor p ϕ q : Y − → V | X | which f actors through the embedding ˆ X ֒ → V | X | , where ˆ X = { ψ ∈ V | X | | ψ : X − ⇀ ◦ G } . Y p ϕ q / / p ϕ q @ @ @ @ @ @ @ @ V | X | ˆ X  ? O O In particu lar , for each T -catego ry X = ( X , a ) we ha ve a : X − ⇀ ◦ X , and therefore obtain the Y oneda functo r y X = p a q : X − → ˆ X . Theor em 1.10. Let ψ : X − ⇀ ◦ Z and ϕ : X − ⇀ ◦ Y be T -modu les. Then, for all z ∈ T Z and y ∈ Y , ~ T p ψ q ( z ) , p ϕ q ( y )  = ( ϕ  ψ )( z , y ) . Pr oo f. First note that the diagra ms V T X × Z 1 T X × p ψ q / / ψ 9 9 s s s s s s s s s s s T X × ˆ X e v O O T X × Z 1 T X × p ψ q / / π 2   T X × ˆ X π 2   Z p ψ q / / ˆ X commute, where the right han d side diagram is ev en a pullback . Then, for z ∈ T Z and y ∈ Y , we ha ve ~ T p ψ q ( z ) , p ϕ q ( y )  = ^ W ∈ T ( T X × ˆ X ) W 7− → T p ψ q ( z ) hom( ξ · T ev( W ) , ϕ ( m X · T π 1 ( W ) , y )) = ^ x ∈ T X ^ X ∈ T T X m X ( X ) = x ^ W ∈ T ( T X × ˆ X ) W 7− → T p ψ q ( z ) , X hom( ξ · T e v( W ) , ϕ ( x , y )) = ^ x ∈ T X ^ X ∈ T T X m X ( X ) = x hom( _ W ∈ T ( T X × Z ) W 7− → z , X ξ · T ψ ( W ) , ϕ ( x , y )) = ^ x ∈ T X hom( _ X ∈ T T X m X ( X ) = x T ξ ψ ( X , z ) , ϕ ( x , y )) = ^ x ∈ T X hom( T ξ ψ · m ◦ X ( x , z ) , ϕ ( x , y )) = ϕ  ( T ξ ψ · m ◦ X )( z , y ) = ( ϕ  ψ )( z , y ) .  Choosin g in particular ψ = a : X − ⇀ ◦ X and Y = G , we obtain the “usual” Y oneda lemma (see als o [CH07]). Cor ollary 1.11. F or eac h ϕ ∈ ˆ X and each x ∈ T X , ϕ ( x ) = ~ T y X ( x ) , ϕ  , that is, ( y X ) ∗ : X − ⇀ ◦ ˆ X is give n by the ev aluation map ev : T X ⊗ ˆ X − → V . As a con sequenc e, y X : X − → ˆ X is fully faithfu l. 10 DIRK HOFMANN 2. C ocomplete T - ca tegories 2.1. T - Cat as an order ed category. W e can transp ort the order -stru cture on hom-se ts from T - Mod to T - Cat via the functor ( − ) ∗ : T - Cat op − → T - Mod , that is, we define f ≤ g whene ver f ∗ ≤ g ∗ . Clearly , we ha ve f ≤ g if and only if g ∗ ≤ f ∗ . W ith this definition w e turn T - Cat into a 2-catego ry , and therefore the (rep resentab le) for getful functor O : T - Cat − → Set fac tors through O : T - Cat − → Ord . As usual, we call T -f unctors f , g : X − → Y equivale nt , and write f  g , if f ≤ g and g ≤ f . Hence, f  g if and only if f ∗ = g ∗ , which in turn is equi v alent to f ∗ = g ∗ . W e call a T - categ ory X L-separat ed (see [HT08] for details) whene ver f  g implies f = g , for all T -functors f , g : Y − → X with codomain X . The T -catego ry V = ( V , hom ξ ) is L -separ ated, and so is each T - categ ory of the for m ˆ X , for a T -catego ry X . T he full subcate gory of T - Cat consis iting of all L-sep arated T -categorie s is denoted by T - Cat sep . A T -catego ry X is called injective if, for all T -fun ctors f : A − → X and fully fai thful T -fun ctors i : A − → B , there exist s a T -fun ctor g : B − → X such that g · i  f . Clearly , for a L -separa ted T -cate gory X we hav e then g · i = f . Lemma 2.1. The following asser tions hol d. (1) Let f , g : X − → Y be T -funct ors between T -cate gories X = ( X , a ) and Y = ( Y , b ) . Then f ≤ g ⇐ ⇒ ∀ x ∈ X . k ≤ b ( e Y ( f ( x ) ) , g ( x )) . In particu lar , for T -functor s f , g : Y − → V | X | we have f ≤ g ⇐ ⇒ ∀ y ∈ Y , x ∈ T X . f ( y )( x ) ≤ g ( y )( x ) . (2) A T -cate gory X is L-separ ated if and only if the underlyin g V -cate gory S X is L-separat ed. (3) W ith X also S X is injective with r espect to fully faithful functo rs, for each T -cate gor y X . Pr oo f. (1) can be fo und in [HT08], (2) foll ows immediat ely from (1), and (3) follo ws from the f acts that S : T - Cat − → V - Cat is actu ally a 2-functor and it’ s left adjoint A : V - Cat − → T - Cat sends fully fait hful V -functo rs to fully faithful T -functors.  One of the most importa nt conce pts in a 2 -cate gory is that of adjointnes s . Here, a T -functo r f : X − → Y is left adjo int if there exists a T -functo r g : Y − → X such that 1 X ≤ g · f and 1 Y ≥ f · g . Passing to T - Mod , f is left adjoin t to g if and on ly if g ∗ ⊣ f ∗ , that is , if and on ly if f ∗ = g ∗ . Bearing in mind Lemma 1.6, we ha v e Pro position 2.2. A T -f unctor f : X − → Y is left adjoint if and only if ther e e xists a function g : Y − → X suc h that f ∗ = g ∗ , that is, b ( T f ( x ) , y ) = a ( x , g ( y ) , for all x ∈ T X and y ∈ Y . 2.2. Cocomplete T -categorie s. Let no w X = ( X , a ) be a T -category . Giv en a T -functor h : Y − → X and a weight ψ : Y − ⇀ ◦ Z in T - Mod , Y ◦ h ∗ / ◦ ψ  X Z ◦ h ∗  ψ ? we call a T -fun ctor g : Z − → X a ψ -weighted colimit of h , and write g  colim( ψ, h ), if g rep resents h ∗  ψ , i.e. if h ∗  ψ = g ∗ . Clearly , if such g e xists, it is unique up to equi v alen ce and t herefor e w e cal l g “the” ψ -weighted colimit of h . W e say that a T -functor f : X − → Y pr eserves the ψ -weighted colimit of h if f · colim( ψ, h )  colim( ψ, f · h ), that is, if ( f · g ) ∗ = ( f · h ) ∗  ψ . A T -functor f : X − → Y is INJECTIVE SP ACES VIA ADJUNCTION 11 cocon tinuous if f preser ves all weighted colimits which exist in X , and a T -c atego ry X is cocomple te if each “weighted diagram” has a colimit in X . A straightfor ward calculati on sho ws that we only nee d to consid er f = 1 X . Lemma 2.3. Let f : Y − → X be a T -fu nctor and ψ : Y − ⇀ ◦ Z be a T -module. Then colim( ψ, f )  colim( ψ ◦ f ∗ , 1 X ) . In particula r , X is cocomplete if and onl y if 1 ∗ X  ψ is r epr ese ntable by some T -fu nctor g : Z − → X , for eac h T -module ψ : X − ⇀ ◦ Z . Furthermor e, a T -functor f : X − → Y is cocontinu ous if and only if f pre serves all ψ -weighted colimits of 1 X . Remark 2.4 . When study ing V -cate gories, one can go ev en one step further and sho w that cocomple teness reduce s to the case Z = G . More precise, a V -categor y X is cocomplete if and on ly if (1 X ) ∗  ψ is repres entable by some V -functor , for each V -module ψ : X − → ◦ G . Ho we ver , for a general theo ry T I am not able to pro ve this. W e let T - Cocont denote the 2-categ ory of all cocomplete T -categor ies and left adjoint T -functors between th em. C orresp onding ly , T - Cocont sep denote s the full sub cate gory of T - Cocont consi sting of all L-separat ed cocomplete T -cate gorie s. Pro position 2.5. The following assertion s hold. (1) Each p ψ q ∈ ˆ X is a colimit of r epr esantables . Mor e pr ecis ely , we have y ∗  ψ = p ψ q ∗ . X ◦ y ∗ / ◦ ψ  ˆ X G ◦ y ∗  ψ ? (2) A left adjoin t T -functor f : X − → Y between T -cate gor ies is cocont inuous. Pr oo f. (1) Let a ∈ T 1 and h ∈ ˆ X . T hen, by Theorem 1.10, ( y ∗  ψ )( a , h ) = ~ T p ψ q ( a ) , h  = p ψ q ∗ ( a , h ) . (2) Let h : A − → X be in T - C at , ψ : A − ⇀ ◦ B in T - Mod , and g  colim( ψ, h ). Then, since f ∗ is a right adjoin t T -modu le, from L emma 1.8 we ded uce ( f · h ) ∗  ψ = f ∗ ◦ ( h ∗  ψ ) = f ∗ ◦ g ∗ = ( f · g ) ∗ .  Theor em 2.6. Let X = ( X , a ) be a T -ca te gory . The following assertio ns ar e equivalent. (i) X is injecti ve. (ii) y X : X − → ˆ X has a left i n verse, i.e . ther e exists a T -functor Sup X : ˆ X − → X such that Sup X · y X  1 X . (iii) y X : X − → ˆ X has a left adjoint Sup X : ˆ X − → X . (i v) X is cocomplet e. Pr oo f. (i) ⇒ (ii) Follo ws immediately fro m the fact that y X : X − → ˆ X is fully fait hful (see Corollary 1.11). (ii) ⇒ (iii) Since Sup X · y X  1 X by hypothes is, it is enough to sho w 1 ˆ X ≤ y X · S up X . Let ψ ∈ ˆ X and x ∈ T X . Then, by Corollary 1.11 and Lemma 2.1, we ha ve ψ ( x ) = ~ T y X ( x ) , ψ  ≤ a ( T (Sup X · y )( x ) , S up X ( ψ )) = a ( x , Sup X ( ψ )) = ~ T y X ( x ) , y X · S up X ( ψ )  = y X · S up X ( ψ )( x ) . 12 DIRK HOFMANN (iii) ⇒ (i v) Assume S up X ⊣ y X and let ψ : X − ⇀ ◦ Y in T - Mod . By Theorem 1.10, for all y ∈ T Y and x ∈ X we ha v e 1 ∗ X  ψ ( y , x ) = ~ T p ψ q ( y ) , y X ( x )  = y ◦ X · p ψ q ∗ ( y , x ) = y ∗ X ◦ p ψ q ∗ ( y , x ) = (Sup X ) ∗ ◦ p ψ q ∗ ( y , x ) = (Sup X · p ψ q ) ∗ ( y , x ) , hence Sup X · p ψ q  colim( ψ, 1 X ). (i v) ⇒ (i) L et i : A − → B be a fully faith ful T -functor . Let f : A − → X be a T -functor . Hence, by cocompl eteness of X , f ∗  i ∗ = g ∗ for some T -functor g : B − → X . Hence ( g · i ) ∗ = g ∗ ◦ i ∗ ≤ f ∗ . On the other hand , from f ∗ = f ∗ ◦ i ∗ ◦ i ∗ we deduce f ∗ ◦ i ∗ ≤ f ∗  i ∗ = g ∗ , hence f ∗ ≤ g ∗ ◦ i ∗ .  Remarks 2.7 . As it hap pens ofte n, the proof of the theorem ab ov e giv es us some further in formation. Firstly , any left in verse S : ˆ X − → X to the Y oneda embedding y X : X − → ˆ X is actu ally left adjoint to y X . I learned thi s us eful f act in the contex t of quant aloid-en riched cate gories from Isar Stubbe. Secondly , the ψ -weighted colimit of 1 X : X − → X in a cocomplete T -cate gory X can be calcula ted as Sup X · p ψ q . Finally , if X is injecti ve , then any T -functor f : A − → X has not only an e xtensio n along a fully faithful T -functor i : A − → B , but e ve n a smallest one with respect to the order on hom-sets in T - C at . Let f : X − → Y be a function . W e define f − 1 : V | Y | − → V | X | to be the mate of the composite | X | ⊗ V | Y | | f |⊗ 1 V | Y | − − − − − − − − − − − → | Y | ⊗ V | Y | e v − − − − − − → V of T -functors. Explicitly , for an y ψ ∈ V | Y | and x ∈ T X , f − 1 ( ψ )( x ) = ψ ( T f ( x ) ). Hence, if f is a T -functo r and ψ ∈ ˆ Y , then f − 1 ( ψ ) = ψ ◦ f ∗ ∈ ˆ X , so hat f − 1 restric ts to a T -funct or f − 1 : ˆ Y − → ˆ X . Theor em 2.8. F or each T -cate gory X , ˆ X is cocomplet e wher e Sup ˆ X = y − 1 X . Pr oo f. Accordi ng to Theo rem 2.6, we ha ve to sho w y − 1 X · y ˆ X = 1 ˆ X . T o do so, let ψ ∈ ˆ X and x ∈ T X . Then, by the Y oneda Lemma (Corollar y 1.11), w e ha ve y − 1 X ( y ˆ X ( ψ ))( x ) = y ˆ X ( ψ )( T y X ( x )) = ~ T y X ( x ) , ψ  = ψ ( x ) , and the asse rtion follows.  Note that th e Theorem abo ve appl ies in particu lar to the discrete T -categ ory X = ( X , e ◦ X ), henc e V | X | is cocompl ete for each se t X . Clearl y , if T 1 = 1, then V | 1 |  V and th erefore th e T -category V is co complete and hence injecti v e in T - Cat . A di ff erent proof of this property of V can be found in [HT08, Lemma 3.18]. Note that also in the proof of [HT08] the conditio n T 1 = 1 is crucial. 2.3. Kan extension. From Theorem 2.6 we know that each T -f unctor f : X − → Y into a cocomple te T -catego ry Y has a smallest extens ion along y X : X − → ˆ X . W e w ill see now that this ext ension is particu larly nice (compare w ith [K el82, Theorem 5.35]) . Theor em 2.9. Composition with y X : X − → ˆ X defines an equiva lence T - Cocont ( ˆ X , Y ) − → T - Cat ( X , Y ) of or der ed sets, for each cocompl ete T -cate gor y Y . That is, for each T -functor f : X − → Y into a cocomple te T -cat e gor y Y , the r e ex ists a (up to equ ivalenc e) unique left adjo int T -fun ctor f L : ˆ X − → Y INJECTIVE SP ACES VIA ADJUNCTION 13 suc h that f L · y X  f ; and, if f ≤ f ′ , then f L ≤ f ′ L . More over , the righ t adjoint to f L is given by p f q ∗ . X y X / / f    > > > > > > > > ˆ X f L ⊣   Y p f q ∗ [ [ Pr oo f. Let f L : ˆ X − → Y be the extensio n of f where ( f L ) ∗ = f ∗  ( y X ) ∗ . Then, by Theorem 1.10, for any p ∈ T ˆ X and y ∈ Y , we ha ve ( f L ) ∗ ( p , y ) = f ∗  ( y X ) ∗ ( p , y ) = ~ p , p f q ∗ ( y )  = p f q ∗ ∗ ( p , y ) , hence f L ⊣ p f q ∗ . Unicity of f L follo ws from Propositi on 2.5. Assume no w f ≤ f ′ . Then f ′ ∗ ≤ f ∗ and therefo re ( f ′ L ) ∗ ◦ ( y X ) ∗ ≤ f ′ ∗ ≤ f ∗ . Hence ( f ′ L ) ∗ ≤ ( f L ) ∗ , that is, f L ≤ f ′ L .  The theore m abov e tel ls us that both inclusion functo rs T - C ocont sep ֒ → T - Cat sep and T - Cocont sep ֒ → T - Cat hav e a left adjoint defined by X 7− → ˆ X which, mo reov er , is a 2-functor . In partic ular , if f : X − → Y is a T -functor , then y Y · f : X − → ˆ Y has a left adjoint extensi on ˆ f : ˆ X − → ˆ Y along y X : X − → ˆ X . X y X / / f   ˆ X ˆ f   Y y Y / / ˆ Y Furthermor e, by Theorem 2.9, the right adjoint of ˆ f is giv en by p ( y Y · f ) q ∗ : ˆ Y − → ˆ X . Explicitly , for each ψ ∈ ˆ Y and each x ∈ T X we ha ve p ( y Y · f ) q ∗ ( ψ )( x ) = ( y Y ) ∗ ◦ f ∗ ( x , ψ ) = ( y Y ) ∗ · T f ( x , ψ ) = ( y Y ) ∗ ( T f ( x ) , ψ ) = ψ ( T f ( x )) , that is, f − 1 = p ( y Y · f ) q ∗ and ˆ f ⊣ f − 1 . Passing to t he under lying ordere d sets , f − 1 : ˆ Y − → ˆ X correspond s to − ◦ f ∗ , th erefore the unde rlying (order -preser ving) map of ˆ f is gi v en by − ◦ f ∗ (see Remark 1 .5). Hence, for ψ ∈ ˆ X and y ∈ T Y we ha ve ψ ◦ f ∗ = ψ ◦ ( f ◦ · b ) = ψ · T f ◦ · T ξ b · m ◦ Y = ψ · T f ◦ · s and ψ ◦ f ∗ ( y ) = _ x ∈ T X ψ ( x ) ⊗ s ( y , T f ( x )) , where b denote s the structure on Y and s = T ξ b · m Y . Consider now the discrete T -cate gory X D = ( X , e ◦ X ). Then, for any T -ca tego ry X , the id entity map j X : X D − → X , x 7− → x is a T -f unctor , and we obtain a left a djoint T -fun ctor b j X : c X D = V | X | − → ˆ X . In th e sequel we find i t con venient to write R X instea d. One easily verifies that its ri ght adjoint j − 1 X : ˆ X − → V | X | is gi ve n by the inclusion map i X : ˆ X ֒ → V | X | . Cor ollary 2.10. F or ea ch T -c ate gory X = ( X , a ) , the inclusion functo r i X : ˆ X − → V | X | has a lef t adjoint given by R X : V | X | − → ˆ X , ψ 7− →         x 7− → _ y ∈ T X ψ ( y ) ⊗ r ( x , y )         , wher e r = T ξ a · m ◦ X . 14 DIRK HOFMANN Cor ollary 2.11. F or eac h function f : X − → Y , the left adjoint to f − 1 : V | Y | − → V | X | is given by V | X | − → V | Y | , ψ 7− →          y 7− → _ x : T f ( x ) = y ψ ( x )          . For a T -functor f : X − → Y , let us write temporari ly f D : ( X , e ◦ X ) − → ( Y , e ◦ Y ) for the same map between the discr ete T -cat egori es. Since ob viously j Y · f D = f · j X , we hav e a commutati v e diagram V | X | c f D / / R X   V | Y | R Y   ˆ X ˆ f / / ˆ Y of T -fun ctors. Furthermore , w e h a ve b f · f − 1 = 1 ˆ X pro vided tha t f is L-de nse, i.e. f ∗ ◦ f ∗ = 1 ∗ X . Satisfying (BC), the functor T : Set − → Set sends surjec tions to surjection s, and therefore each surjecti ve T -fun ctor f is L-dense. 2.4. Cocomplete T -cate gories as E ilenberg –Moor e algebras. Pro position 2.12. Let f : X − → Y be a T -functor between cocomple te T -cate gories. Then the followin g assert ions ar e equivalen t. (i) f is left adjo int. (ii) f is coc ontinu ous, that is, f pr eserves all wei ghted colimits. (iii) W e have f · Sup X  S up Y · ˆ f , wher e Sup X ⊣ y X and Sup Y ⊣ y Y . ˆ X ˆ f / / Sup X    ˆ Y Sup Y   X f / / Y Pr oo f. The implica tion (i) ⇒ (ii) we pro ve d already in Propositio n 2.5 . T o see that (ii) ⇒ (iii), recall that Sup X  colim(( y X ) ∗ , 1 X ) and therefo re f · Sup X  colim(( y X ) ∗ , f ). W ith the help of Lemma 1.8, we get ( f · Sup X ) ∗ = f ∗  ( y X ) ∗ = ( y ∗ Y ◦ ( y Y · f ) ∗ )  ( y X ) ∗ = y ∗ Y ◦ (( y Y · f ) ∗  ( y X ) ∗ ) = y ∗ Y ◦ ˆ f ∗ = (Sup Y · ˆ f ) ∗ . Finally , to obtain (iii) ⇒ (i), we sho w that f ⊣ Sup X · f − 1 · y Y . In fact, (Sup X · f − 1 · y Y ) ∗ = y ∗ Y ◦ f − 1 ∗ ◦ Sup ∗ X = S up Y ∗ ◦ ˆ f ∗ ◦ Sup ∗ X = f ∗ ◦ Sup X ∗ ◦ Sup ∗ X = f ∗ ◦ y ∗ X ◦ Sup ∗ X = f ∗ .  Example 2.13. Recall from S ubsecti on 2.9 that, for each T -fun ctor f : X − → Y , we hav e an adjunction ˆ f ⊣ f − 1 in T - Cat . T he underlying (order -preserving) m aps of ˆ f and f − 1 are giv en by − ◦ f ∗ and − ◦ f ∗ respec tiv ely . Furthe rmore, we ha ve ˆ ˆ f ⊣ d f − 1 . Since y Y · f = ˆ f · y X , we ob tain b y Y · ˆ f = ˆ ˆ f · b y X and the refore y − 1 X · d f − 1 = f − 1 · y − 1 Y . Hence, by Theorem 2.8 and Propos ition 2.12, f − 1 has a right adjo int f • : ˆ X − → ˆ Y in T - Cat . The underly ing order- preservi ng map of f • we iden tified in Remark 1.5 as ( − )  f ∗ . The pair of adjoin t functors T - Cocont sep ֒ → T - Cat sep and c ( − ) : T - Cat sep ֒ → T - Cocont sep induce s monad on T - Cat sep , denoted as I = ( c ( − ) , y , µ ). By Theorem 2.9, we hav e that f ≤ g implies ˆ f ≤ ˆ g , so that c ( − ) is a 2-functo r . Furthermore, since obvi ously y ˆ X · y X = y ˆ X · y X , we ha ve ( y ˆ X ) ∗ ≤ ( b y X ) ∗ , that is, b y X ≤ y ˆ X . In general, a monad S = ( S , d , l ) on a locally thin 2-cate gory X is of Ko ck -Z ¨ oberl ein type (see [K oc95]) if S is a 2-functor and S d X ≤ d S X , for all X ∈ X . In fact, in [K oc95] it is sho wn that INJECTIVE SP ACES VIA ADJUNCTION 15 Theor em 2.14. Let S = ( S , d , l ) be a monad on a loca lly thin 2-cate gory X wher e S is a 2- functor . Then the foll owing assertions ar e equivalen t. (i) S d X ≤ d S X for all X ∈ X . (ii) S d X ⊣ l X for all X ∈ X . (iii) l X ⊣ d S X for all X ∈ X . (i v) F or all X ∈ X , a X -morphism h : S X − → X is the structu r e m orphis m of a S -alg ebra if and only if h ⊣ d X with h · d X = 1 X . The consid eration s abov e tell us tha t the monad I = ( c ( − ) , y , µ ) on T - Cat sep is of K ock- Z ¨ oberle in typ e. Furthermor e, by Theorem 2.6 and Proposition 2.12 we hav e Theor em 2.15. ( T - Cat sep ) I  T - Cocont sep . Hence, in particu lar , T - Cocont sep is complet e. Theorem 2.14 also help s us to compute the multip lication µ of I : for an y (L-separated) T -cat egor y X we ha v e b y X ⊣ µ X and b y X ⊣ y − 1 X , henc e µ X = y − 1 X . 2.5. Example: topological spaces. W e consider no w T = U 2 = ( U , 2 , ξ 2 ). Hence T - Cat = T op is the category of topolog ical spaces and continuou s maps, and T - Cat sep = T op 0 its full subcate gory of T 0 -space s (see also [CH07, HT08]). Then M ( X ) = ( U X , ≤ ) is the order ed set wit h x ≤ y ⇐ ⇒ { A | A ∈ x } ⊆ y , and the topo logy on | X | is giv en by the Zariski-clo sure define d by x ∈ cl A : ⇐ ⇒ \ A ⊆ x ⇐ ⇒ x ⊆ [ A . In [HT08] we observ ed already that the down-c losure as w ell as the up-clo sure of a Zariski-close d set is again Z ariski- closed. A pres heaf ψ ∈ ˆ X can be iden tified with th e Zariski -closed and d own -closed subset A = ψ − 1 (1) ⊆ U X , and we consider ˆ X = {A ⊆ U X | A is Zariski -closed and down-clo sed } . The topolo gy on ˆ X is the compact-o pen topo logy , which has as basic open sets B ( B , { 0 } ) = {A ∈ ˆ X | A ∩ B = ∅ } , B ⊆ U X Zariski- closed. The Y oneda map y X : X − → ˆ X is gi ven by y X ( x ) = { x ∈ U X | x → x } . Fo r x ∈ U X , U y X ( x ) is the ultrafilte r generated by the sets {{ a | a → x } | x ∈ A } ( A ∈ x ) , and the Y oneda lemma (Corollar y 1.11 ) states that it con ver ges to A ∈ ˆ X precisely if x ∈ A . W e hav e maps Φ X : P ( U X ) − → F X , A 7− → \ A and Π X : F X − → P ( U X ) , f 7− → { x ∈ U X | f ⊆ x } . where P ( U X ) den otes the p ower set of U X and F X the set of all (possib ly improper) filters on X . C learly , we ha ve f = Φ X ( Π X ( f )) and A ⊆ Π X ( Φ X ( A )) for f ∈ F X and A ∈ P ( U X ) . Furthermore, A = Π X ( Φ X ( A )) if and only if A is Z ariski- closed. W e let F 0 X denote the set of all filters on the lattice τ of open sets of a topological space X , and F 1 X the set of all filters on the lattice σ of closed sets of X . For each filter f on X we can consider f ∩ τ ∈ F 0 X and f ∩ σ ∈ F 1 X , and f is det ermined by this restriction precisely if f has a basis of open respecti vel y closed sets. In [HT 08] we sho wed that f = T A has a basis of open sets if and only if A is down-cl osed, and f has a basis of closed sets if and only if A is up-clos ed. Hence ˆ X  F 0 X and {A ⊆ U X | A is Zariski- closed and up-closed }  F 1 X , 16 DIRK HOFMANN and the first homeomorphi sm we also denote as Φ X : ˆ X − → F 0 X , A 7− → ( T A ) ∩ τ . Let B ( B , { 0 } ) be a basic open set of the topology of ˆ X . Since B ( B , { 0 } ) = B ( ↑B , { 0 } ), we can assume that B is up-closed . Hence, unde r the bijections abov e, F 0 ( X ) has { f ∈ F 0 ( X ) | ∃ A ∈ f , B ∈ g . A ∩ B = ∅ } ( g ∈ F 1 ( X )) as basic open sets. C learly , it is enough to conside r g =  B the princi pal filter induced by a close d set B , so that all sets { f ∈ F 0 ( X ) | ∃ A ∈ f . A ∩ B = ∅ } = { f ∈ F 0 ( X ) | X \ B ∈ f } ( B ⊆ X closed) form a basis for the topology on F 0 ( X ). W e hav e sho wn that our p resheaf space ˆ X is homeomorphi c to the filter space F 0 ( X ) consid ered in [Esc97]. Furthermore , for a c ontinuo us m ap f : X − → Y , f − 1 : ˆ Y − → ˆ X corres ponds to f − 1 : F 0 Y − → F 0 X , g 7− → { f − 1 ( B ) | B ∈ g } in the sense that the diagram ˆ Y Φ Y / / f − 1   F 0 Y f − 1   ˆ X Φ X / / F 0 X commutes. Hence, since ˆ f ⊣ f − 1 as well as F 0 f ⊣ f − 1 , Φ = ( Φ X ) X is a natura l isomorphism from c ( − ) : T op 0 − → T op 0 to F 0 : T op 0 − → T op 0 . Since Φ X ( y ( x )) = { U ∈ τ | x ∈ U } is the neighborh ood filter of x ∈ X , the monad I = ( c ( − ) , y , y − 1 ) is isomorp hic to the fi lter monad on T op 0 consid ered in [Esc9 7]. 2.6. Cocomplete T -categorie s are algebras ov er Set and V - Cat sep . W e are now aiming to pro v e that the for get ful funct or G : T - Cocont sep − → Set is monadi c. Clearly , G has a left adjoint giv en by the composite Set discrete − − − − − − − − − − → T - Cat sep c ( − ) − − − − − − − → T - Cocont sep . Furthermor e, w e ha ve the follo w ing elementary facts. Lemma 2.16. Let f : X − → Y and g : Y − → X be T -functors with f ⊣ g wher e X , Y ar e L-separ ated. (1) The following assertio ns ar e equivalent . (i) f is an epimor phism in T - Cat sep . (ii) f · g = 1 Y . (iii) f is a split epimorphis m in T - Cat sep . (2) The following assertio ns ar e equivalent . (i) f is a mono morphism in T - Cat sep . (ii) g · f = 1 X . (iii) f is a split monomorphis m in T - Cat sep . Pr oo f. From f ⊣ g we obtain f · g · f = f . If f is an epi morphism in T - Cat sep , then f · g = 1 Y ; if f is a monomorph ism in T - Cat sep , then g · f = 1 X .  Cor ollary 2.17. G r eflects isomorphi sms. Pr oo f. If f : X − → Y in T - Cocont sep is bijecti ve, then f is an isomorphism in T - Cat sep and therefore also in T - Cocont sep .  INJECTIVE SP ACES VIA ADJUNCTION 17 In order to conclud e that G is m onadic , it is left to sho w that T - Cocont sep has and G prese rves co- equali ser of G -equi v alence relations (s ee, for instan ce, [MS04, Corollary 2.7]). Hence, let π 1 , π 2 : R ⇒ X in T - Cocont sep be an equi v alence relatio n in Set , where π 1 and π 2 are th e proj ection maps. Let q : X − → Q be its coeq ualiser in T - Cat . The follo wing fact will be cruci al in the sequel: ( ‡ ) ˆ R b π 1 / / b π 2 / / ˆ X ˆ q / / ˆ Q is a split fork in T - Cat sep . The splitting here is gi ve n by q − 1 : ˆ Q − → ˆ X and π − 1 1 : ˆ X − → ˆ R . First note tha t, since both π 1 and q are surjec tiv e, we hav e ˆ q · q − 1 = 1 and b π 1 · π − 1 1 = 1. Hence, in order to obtain ( ‡ ), we need to sho w q − 1 · ˆ q = b π 2 · π − 1 1 . Note that we ha ve ˆ q = ˆ q · b π 1 · π − 1 1 = ˆ q · b π 2 · π − 1 1 , and there fore q − 1 · ˆ q = q − 1 · ˆ q · b π 2 · π − 1 1 ≥ b π 2 · π − 1 1 . W e will giv e a proof for ( ‡ ) at the end of this subsection , and sho w fi rst how ( ‡ ) can be used to prov e monadic ity of G . Observe first that, being a split fork, ˆ R b π 1 / / b π 2 / / ˆ X ˆ q / / ˆ Q is a coequali ser diagram in T - Cat and T - Cat sep . Hence, there is a T -func tor Sup Q : ˆ Q − → Q with Sup Q · ˆ q = q · Sup X and Sup Q · y Q = 1 Q . The situation is depicted belo w . R π 1 / / π 2 / / y R   X q / / y X   Q y Q   1 Q z z ˆ R b π 1 / / b π 2 / / Sup R   ˆ X ˆ q / / Sup X   ˆ Q Sup Q   R π 1 / / π 2 / / X q / / Q W e conc lude that Q is L -separ ated and coco mplete, and q : X − → Q is cocontinuo us. Next we sho w tha t R π 1 / / π 2 / / X q / / Q is indee d a coqualiser diagram in T - Cocont sep . Note that ˆ R b π 1 / / b π 2 / / ˆ X ˆ q / / ˆ Q is a coeq ualiser diagram in T - Cocont sep since c ( − ) : T - Cat − → T - Coc ont sep is left adjoint. Let h : X − → Y be a coconti nuous T -fun ctor with cocomplete codomain such that h · π 1 = h · π 2 . Then there e xists a cocon tinuous T -fu nctor f : ˆ Q − → Y su ch that f · ˆ q = h · S up X . W e consi der now f · y Q : Q − → Y . Then f · y Q · q = f · ˆ q · y X = h · Sup X · y X = h . 18 DIRK HOFMANN Furthermor e, Sup Y · ˆ f · c y Q · ˆ q = f · Sup ˆ Q · c y Q · ˆ q = f · ˆ q (Sup ˆ Q = µ Q the multiplic ation of th e monad I ) = h · Sup X = f · y Q · q · Sup X = f · y Q · S up Q · ˆ q , and therefo re S up Y · [ f · y Q = f · y Q · S up Q , i.e. f · y Q is cocon tinuou s. Remark 2.1 8 . Being c ocontin uous, f · y Q is left adjoin t. In f act, o ne can directly sho w f · y Q ⊣ q · l , where l : Y − → X is right adjoint to h : X − → Y . T o do so, let g : Y − → ˆ Q be right adjoi nt to f : ˆ Q − → Y . Then y X · l = q − 1 · g , and therefore g = ˆ q · y X · l and l = Sup X · q − 1 · g . Hence, we ha ve f · y Q · q · l = f · ˆ q · y X · l = f · g ≤ 1 Y and q · l · f · y Q = q · Sup X · q − 1 · g · f · y Q ≥ q · Sup X · q − 1 y Q = S up Q · ˆ q · q − 1 y Q = 1 Q . Finally , we pro ve ( ‡ ). Let π 1 , π 2 : R ⇒ X be an equiv ale nce relation in Set , and q : X − → Q its quotie nt. W e typically write x ∼ x ′ for ( x , x ′ ) ∈ R . Furthermore, for x , x ′ ∈ T X we write x ∼ x ′ whene ve r the pair ( x , x ′ ) belong s to the kernel relation of T q . Since T has (BC), we ha ve x ∼ x ′ ⇐ ⇒ ∃ w ∈ T R . ( T π 1 ( w ) = x ) & ( T π 2 ( w ) = x ′ ) . Furthermor e, we ha v e to warn the reade r that, when t alking abou t an e qui v alence relatio n π 1 , π 2 : R ⇒ X in T - C at or T - Cat sep , w e always include that the canonical map R ֒ → X × X is an embe dding (and no t ju st a monomorph ism). Clearly , a sub- T -categor y R ֒ → X × X is an equiv alence relatio n in T - Cat respecti vely in T - Cat sep if and only if it is an equi v alen ce relatio n in Set . Lemma 2.19. L et X = ( X , a ) be a L-separ ated T -cate gory and π 1 , π 2 : R ⇒ X be an equiva lence r elati on in T - Cat sep . In add ition, assu me that π 2 ⊣ ρ 2 1 . Then, for all x , x ′ ∈ T X with x ∼ x ′ and all x ′ ∈ X , ther e e xists x ∈ X suc h that x ∼ x ′ and a ( x ′ , x ′ ) ≤ a ( x , x ) . Pr oo f. Since π 2 is surjecti ve , we hav e π 2 · ρ 2 = 1 X . Let w ∈ T R such that T π 1 ( w ) = x and T π 2 ( w ) = x ′ . Then a ( x ′ , x ′ ) = a ( T π 2 ( w ) , x ′ ) = a × a ( w , ρ 2 ( x ′ )) ( ρ 2 ( x ′ ) = ( x , x ′ ) for some x ∼ x ′ ) = a ( x , x ) ∧ a ( x ′ , x ′ ) , hence a ( x ′ , x ′ ) ≤ a ( x , x ).  1 Note that, since R is symmetric, π 1 is left adjoint precisely if π 2 is so. INJECTIVE SP ACES VIA ADJUNCTION 19 Our next goal is to describe the quotien t q : X − → Q of π 1 , π 2 : R ⇒ X in T - Cat . In gener al, the quotient structure in T - Cat is di ffi cult to handle , see [Hof05] for details. The situation is much better in T - Gph , the category of T -graphs and T -g raph m orphis ms. Here a T -g raph is a pair ( X , a ) consis ting of a set X and a T -relation a : X − ⇀ 7 X satisfyin g e ◦ X ≤ a , and T -gra ph morph isms are defined as T -fu nctors. Clearly , we ha ve a full embedding T - Cat ֒ → T - Gph . A surjecti v e T -gr aph morp hism f : ( X , a ) − → ( Y , b ) is a quotie nt in T - Gph if and onl y if b = f · a · T f ◦ (see also [CH03]), and the full embed ding T - Cat ֒ → T - Gph reflects quot ients. Furthe rmore, we call a T -gra ph morphism (or a T -functor) f pr oper if b · T f = f · a (see [CH04]). One easily veri fies that, if f : X − → Y is a proper surjec tion, then f is a quotien t in T - Gph , and with X als o Y is a T -cate gory . Cor ollary 2.20. Conside r the same situatio n as in the lemma abo ve. Let q : X − → Q be the quotie nt of π 1 , π 2 : R ⇒ X in T - Gph . Then q is pr oper , and ther efor e Q is a T -cate gory and q : X − → Q is the quotie nt of π 1 , π 2 : R ⇒ X in T - Cat . Pr oo f. Let x ∈ T X and y ∈ Q , i.e. y = q ( x ) for some x ∈ X . Wi th c d enoting the n structu re on Q , we ha ve c ( T q ( x ) , y ) = _ { a ( x ′ , x ′ ) | x ′ ∼ x , x ′ ∼ x } = _ { a ( x , x ′ ) | x ′ ∼ x } = _ { a ( x , x ′ ) | x ′ ∈ X , q ( x ′ ) = y } .  Cor ollary 2.21. W ith the same nota tion as above , M ( q ) : M ( X ) − → M ( Q ) is pr oper . Pr oo f. Just observ e that both diagrams T X _ m ◦ X   T q / / T Q _ m ◦ Q   T T X T T q / / _ T ξ a   T T Q _ T ξ c   T X T q / / T Q are c ommutati ve: the upper one si nce m has (BC), t he lo wer on e since q is pr oper and T ξ is a fu nctor .  W e are no w in the position to show ( ‡ ). Let π 1 , π 2 : R ⇒ X in T - Cocont sep be an equi v alen ce rela tion in Set . Note that R ֒ → X × X is left adjoint and injecti ve, hence a split m onomorp hism and theref ore an embeddi ng in T - Cat sep . Hence, by Corollary 2.20 , its quotient q : X − → Q in T - Cat is proper , and so is M ( q ) : M ( X ) − → M ( Q ) by C orolla ry 2.21. Let ψ ∈ ˆ X and x ∈ T X . The structure on X and Q we denote as a and c resp ecti vel y , and put r = T ξ a · m ◦ X and s = T ξ c · m ◦ Q . W e ha ve ( q − 1 · ˆ q ( ψ ))( x ) = ˆ q ( ψ )( T q ( x )) = _ x ′ ∈ T X ψ ( x ′ ) ⊗ s ( T q ( x ) , T q ( x ′ )) = _ ( x ′ ∈ T X ) _ ( x ′′ : x ′′ ∼ x ′ ) ψ ( x ′ ) ⊗ r ( x , x ′′ ) and ( b π 2 · π − 1 1 ( ψ ))( x ) = _ ( x ′ ∈ T X ) _ ( w : T π 2 ( w ) = x ′ ) ψ ( T π 1 ( w )) ⊗ r ( x , x ′ ) = _ ( x ′ ∈ T X ) _ ( x ′′ : x ′′ ∼ x ′ ) ψ ( x ′′ ) ⊗ r ( x , x ′ ) . W e conclud e q − 1 · ˆ q = b π 2 · π − 1 1 . 20 DIRK HOFMANN Theor em 2.22. The for getf ul functor G : T - Cocont sep − → Set is monadic. As a consequ ence , T - Cocont sep is cocomp lete. Theor em 2.23. The for getfu l functor S : T - Cocont sep − → V - Cat sep is monadic. Pr oo f. Clearly , S has a left adjoint and reflects isomorphisms . W e sho w that S preserv es coequal is- ers of S- contract ible equiv alence relations (see [MS04, Theor em 2.7]). Hence, let π 1 , π 2 : R ⇒ X in T - Cocont sep be a contract ible equi v alenc e relation in V - C at sep . Then π 1 , π 2 : R ⇒ X is also an equi v a- lence relation in Set , and hen ce its coequaliser q : X − → Q in Set underlies its coequaliser q : X − → Q in T - Cocont sep , moreo ver , q : X − → Q is a prop er T -f unctor . Consequ entely , the underlyin g V -functor q : X − → Q is prope r as well, and therefore a coequalis er of π 1 , π 2 : R ⇒ X in V - Cat sep .  2.7. Densely injectiv e T -ca tegories . Another well-kno wn result in T opology is Theor em 2.24. The algebr as for the pr ope r filter monad on T op 0 ar e pr ecisel y the T 0 -space s which ar e injecti ve w ith r espec t to dense embeddings. In the langu age of con v er gence, a con tinuou s map f : X − → Y is dense whenev er ∀ y ∈ Y ∃ x ∈ T X . U f ( x ) → y , and we obse rve that U f ( x ) → y ⇐ ⇒ x f ∗ y . This suggests the follo wing Definition 2.25. A T -module ϕ : X − ⇀ ◦ Y is called inhabit ed if k ≤ ^ y ∈ Y _ x ∈ T X ϕ ( x , y ) . A T -functor f : X − → Y is called dense if f ∗ is inhab ited. W e hasten to remark that f ∗ is inhab ited, for each T -fu nctor f : X − → Y . Hence Pro position 2.26. E ach l eft adjoint T -functor is dense. By definit ion, ϕ : X − ⇀ ◦ Y is inhabited if and on ly if k ≤ ϕ ◦ k , where k denotes the co nstant T -relation k : T 1 × Z − → V with valu e k ∈ V , for a set Z . Consequentel y , with ϕ : X − ⇀ ◦ Y and ψ : Y − ⇀ ◦ Z also ψ ◦ ϕ is inhabite d. Furthermore, if ϕ is in habited and ϕ ≤ ϕ ′ , th en ϕ ′ is in habited too. Note also tha t each surjec tiv e T -fu nctor is dense. Pro position 2.27. C onside r the (up to  ) commutative triangle X f   g    ? ? ? ? ? ? ? Y h / / Z of T -functor s. T hen the following assert ions hold. (1) If h and f ar e de nse, then so is g. (2) If g is dense and h is fully faithful , then f is dense. (3) If g is dense , then h is dense. Pr oo f. (1) i s ob vious si nce inhab ited T -modules compos e. T o see (2), no te that f rom h ∗ ◦ f ∗ = g ∗ follo ws f ∗ = h ∗ ◦ g ∗ , henc e f ∗ is inhab ited and therefore f is dense. (3) can be sho wn in a similar way .  INJECTIVE SP ACES VIA ADJUNCTION 21 By the Y oneda Lemma (Corollary 1.11), for each ψ ∈ ˆ X we hav e _ x ∈ T X ( y X ) ∗ ( x , ψ ) = _ x ∈ T X ψ ( x ) . Hence, with X + = { ψ ∈ ˆ X | ψ is inhabited } and the structure bein g inherited from ˆ X , the restriction y X : X − → X + of the Y oneda embe dding is dense. Furthermore, for a T - module ϕ : X − ⇀ ◦ Y we ha ve ϕ is inhab ited ⇐ ⇒ p ϕ q : Y − → ˆ X fac tors through X + ֒ → ˆ X . W e call a T -category X densely injecti ve if, for all T -functors f : A − → X and ful ly faithfu l and dense T -functors i : A − → B , there exists a T -fun ctor g : B − → X such that g · i  f . A T -cate gory X is called inhab ited-coc omplete if X has all ϕ -weighted co limits where ϕ is inhabite d. Note that, when passing from A f / / ◦ ϕ  X B to X 1 X / / ◦ ϕ ◦ f ∗  X , B with ϕ also ϕ ◦ f ∗ is inhabited, so that it is enough to consider f = 1 X in the definition of inhabited- cocompl ete. A T -fun ctor f : X − → Y is inhabite d-cocon tinuous if f preser ves all ϕ -weighted col imits where ϕ is inha bited. Let T - ICocont denote the category of inhab ited-co complete T -cate gorie s and inhabi ted-coc ontinuous T -functors between them, and T - ICocont sep denote s its full subca tego ry of L- separa ted T -cate gorie s. Lemma 2.28. F or each T -cate gory X , X + is closed under inhabite d colimits in ˆ X . In particul ar , X + is inhab ited-coc omplete . Pr oo f. W e consider the diagram X + ι / / ◦ ϕ  ˆ X , Y with ι : X + ֒ → ˆ X being the inclusi on functor and ϕ inhabi ted. Its colimit in ˆ X is gi ven by y − 1 X · p ϕ ◦ ι ∗ q : Y − → ˆ X . Hence, for an y y ∈ Y and x ∈ T X , y − 1 X · p ϕ ◦ ι ∗ q ( y )( x ) = ϕ ◦ ι ∗ ( T y X ( x ) , y ) ≥ ϕ · T ι ◦ ( T y X ( x ) , y ) = ϕ ( T y X ( x ) , y ) = ϕ ◦ ( y X ) ∗ ( x , y ) , where in the las t two expessio ns we consider y X : X − → X + . Since ϕ ◦ ( y X ) ∗ is inhabite d, the T -fun ctor y − 1 X · p ϕ ◦ ι ∗ q : Y − → ˆ X takes v alues in X + and the asserti on follows.  From the observ ation s made so far it is no w clear that we ha v e the same series of results for den sely injecti ve and inhabited-c ocomplete T -cate gorie s as we prov ed for injecti v e and c ocomplete T -categor ies. Theor em 2.29. Let X be T -cate gory . (1) Each ψ ∈ X + is an inhab ited colimit of r epr ese ntables. (2) The following assertio ns ar e equivalent . (i) X is densel y injective . 22 DIRK HOFMANN (ii) y X : X − → X + has a left in verse S up + X : X + − → X . (iii) y X : X − → X + has a left adjoin t Sup + X : X + − → X . (i v) X is inhabited -cocomplet e. (3) Compositio n with y X : X − → X + define s an equivalenc e T - ICocont ( X + , Y ) − → T - Cat ( X , Y ) of or der ed sets, for each inhabi ted-coc omplete T -cate gory Y . W e ha ve just seen that the inclus ion functor T - ICocont sep ֒ → T - Cat sep has a left adjoin t ( − ) + : T - Cat sep − → T - ICocont sep . In fact, sinc e fo r each T -fun ctor f : X − → Y and each ψ ∈ X + we hav e ˆ f ( ψ ) ∈ Y + , the T -functor f + : X + − → Y + is just the restriction of ˆ f to X + and Y + . W ith a similar proof as for Proposit ion 2.12 one shows Pro position 2.30. Let f : X − → Y be a T -fun ctor between inh abited- cocomplete T -cate gories. Then the followin g assertions ar e equivalen t. (i) f is inh abited-c ocontinuous. (ii) W e have f · Sup + X  S up + Y · ˆ f . X + f + / / Sup + X    Y + Sup + Y   X f / / Y The induced monad on T - Cat sep we denote as I + = (( − ) + , y , µ ). W ith the same argumen ts used in 2.4 one ver ifies that I + is of K ock-Z ¨ oberlein type. W e conclude Theor em 2.31. T - ICocont sep  ( T - Cat sep ) I + . Finally , w e consid er a T -functor f : X − → Y . Then ˆ f : ˆ X − → ˆ Y has a right adjoin t f − 1 : ˆ Y − → ˆ X gi ven by f − 1 ( ψ ) = ψ ◦ f ∗ . Clearl y , if f is dens e, then f − 1 can be restricted to f − 1 : Y + − → X + and we ha ve f + ⊣ f − 1 . In particula r , y + X : X + − → X ++ is left adjoint to y − 1 X : X ++ − → X + , which tells us that the multiplic ation µ X of I + is also gi ve n by y − 1 X . Pro position 2.32. T he follo wing are e quivale nt for a T -functor f : X − → Y . (i) f is den se. (ii) f + is left adjoint. (iii) f + is dense . If f is a inhabit ed-coco ntinuous T -functor between inhabite d cocomplete T -ca te gories, then an y of the condit ions above is equival ent to (i v) f is left adjoin t. Pr oo f. The implicatio n (i) ⇒ (ii) we pro ved abov e, (ii) ⇒ (iii) and (iv) ⇒ (i) foll ow from Propos ition 2.26 and ( iii) ⇒ (i) from Proposition 2.27. Finally , (ii) ⇒ (iv) can be s ho wn as (iii) ⇒ (i) o f Prop osition 2.12.  Finally , thanks to the considerat ions made abo ve, also R + π + 1 / / π + 2 / / X + q + / / Q + is a split fork in T - Cat sep . Consequente ly , with the same pro of as in 2.6, we conclude that the forget ful functo r T - ICocont sep − → Set i s monadic. INJECTIVE SP ACES VIA ADJUNCTION 23 Remark 2.33 . The results of this subsec tion su ggest that in the future one should conside r cocompletene ss with respect to a class Φ of T -modules, i.e. use [KS05]. Besides the class es con sidered in this pap er , anothe r reasonable choice is Φ being the class of al l right adjoint T -mo dules. In fact, this case is stud ied in [CH07, HT08] wher e the Φ -cocompl ete T -catego ries are called L-complete (resp. Cauchy-c omplete). Furthermor e, it is easy to see that any T -fun ctor prese rves colimits inde xe d by a right adjoin t weight, so th at the categor y of L-separat ed and Φ -cocomplete T -categor ies and Φ -coco ntinuou s T -fun ctors is precis ely the full subcate gory T - Cat cpl of L-comp lete an d L-separated T -ca tegor ies of T - Cat . But be awa re that, thought with the same tech niques we obtain mona dicity of T - Cat cpl ov er T - Cat sep , the proof in 2.6 doe s not work here. The problem is that the T - functor q − 1 : ˆ Q − → ˆ X doe s not restrict to ˜ Q and ˜ X 2 since q ∗ is in genera l not right adjoint. This is not a surprise, since, for instanc e, an y ordered set is L-complete , hen ce t he categor y of L -complete and L-separated ordered s et coincides w ith the categor y o f anti-sy mmetric ordered sets (and monoto ne maps). The canoni cal forget ful functor from this cate gory to Set is s urely not mo nadic. Furthermor e, the canon ical forge tful functor from th e cate gory of L-c omplete and L-separat ed topological spaces ( = sober spaces) and continuous m aps to Set i s also not monadic. R eference s [AHS90] Jiˇ r ´ ı Ad ´ amek, Horst Herrlich, and George E . Strecker . Abstract and concr ete cate gories . P ure and Applied Mathe- matics (New Y ork). John Wile y & Sons Inc., Ne w Y ork, 1990. The joy of cats, A W iley-In terscience Publication. [Bar70] Michael Barr . Relational algeb ras. In Reports of the Midwest Ca te gory Semina r , IV , pages 39 –55. Lecture Notes in Mathematics, V ol. 137. Springer, Berlin, 1970. [BCSW83] Renato Betti , Aurelio Carbon i, Ross Street, and Robert W alters. V ariation through enrichment. J . Pur e Appl. Alg e- bra , 29(2 ):109–127, 1983. [Ben63] J. Benabou. Categories av ec multiplication. C. R. Acad. Sci., P aris , 256:1887–18 90, 1963. [Ben65] J. Benabou. Categories r ´ elativ es. C. R. Acad. Sci., P aris , 260:38 24–3827, 1965. [Bur71] Albert Burroni. T -cat ´ egories (cat ´ egories dans un triple). Cahiers T opolog ie G ´ eom. Di ff ´ er entielle , 12:215–321, 1971. [BvBR98] M. M. Bonsangue , F . van Breugel, and J. J. M. M. Rutten. Generalized metri c spaces: completion, topology , and po werdomains via the Yoneda embedding. Theor et. Comput. Sci. , 193(1-2):1–51, 1998. [CH03] Maria Manuel Clementino and Dirk Hofmann. T opological features of l ax algebras. Appl. C ate g. Structur es , 11(3):267–2 86, 2003. [CH04] Maria Manuel Clementino and Dirk Hofmann. E ff ectiv e descent morphisms i n categories of lax algebras. Appl. Cate g. Structur es , 12(5-6):413–4 25, 2004. [CH07] Maria Manuel Clementino and Dirk Hofmann. Lawvere completeness in Topolo gy . T echnical report, Uni v ersity of A veiro, 2007 , arXiv :math.CT / 0704.3976. [CHT04] Maria Manuel Clementino, Dirk Hofmann, and W alter Tholen. One setting for al l: metric, topology , uniformity , approach structure. Appl. Cate g. Structur es , 12(2):127–15 4, 2004. [CT03] Maria Manuel C lementino and W alter Tholen. Metric, topology and multicategory—a common approach. J. Pur e Appl. Algebra , 179(1-2):13–47, 2003. [Day75] Alan Day . Fil ter monads, con tinuous latt ices and closure systems. Can. J . Math. , 27:50–59, 1975. [EK66] Samuel Ei lenberg and G. M ax Kelly . Closed categories. In P r oc. Conf. C ate gorical Algebra (La Jolla, Calif., 1965) , pages 421–562 . Springer , New Y ork, 1966. [Esc97] Mart ´ ın H ¨ otzel E scard ´ o. Injectiv e spaces via t he fil ter monad. I n P r oceeding s of the 12th Summer Confer ence on Genera l T opology and its Applications (North Bay , ON, 1997) , volume 22 , pages 97–100, 1997. [Hof05] Dirk Hofma nn. An algebraic description of re gular e pimorphisms in topology . J. Pur e App l. Alg ebra , 199(1-3):71– 86, 2005. [Hof07] Dirk Hofmann. T opological theories and closed objects. Adv . Math. , 215(2):789–8 24, 2007. [HT08] Dirk Hofmann and W alter Tholen. Lawve re completion and sep aration via closure. T echnical report, Uni versity of A veiro, 2008 , arXiv :math.CT / 0801.0199. 2 Here ˜ X = { ψ ∈ ˆ X | ψ right adjoint } . 24 DIRK HOFMANN [Kel8 2] Gre gory Maxwell Kelly . B asic concepts of enriched catego ry theory , volume 64 of London Mathematical Society Lectur e Note Series . Cambridge Univ ersity Press, Cambridge, 1982. [K oc95] And ers Kock. Mo nads for which structures are adjoint to units. J. Pur e Appl. Algebr a , 104(1):41–5 9, 1995. [KS05] Gregory Maxwell Kelly and V incent Schmitt. Notes on enriched catego ries with colimit s of some class. Theory Appl. Cate g. , 14:3 99–423, 2005. [Law73] F . W illiam Lawve re. M etric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano , 4 3:135– 166 (1974), 1973. Also in: Repr . Theory Appl. Categ. 1:1–3 7 (electronic), 2002. [Low9 7] Robe rt Lo wen. A ppr oac h spaces . Oxford Mathematical Mo nographs. T he Clarendon Press Oxford Uni v ersity Press, Ne w Y ork, 1997. The missing link in the topology-uniformity-metric triad, Oxford Science Publications. [MS04] John MacDonald and M anuela Sobral. Asp ects of monads. In Catego rical foun dations , v olume 97 of Encyclopedia Math. Appl. , pages 213–26 8. Cambridge Univ . Press, Cambridge, 2004. [Sco72] Dana Scott. Continuous lattices. In T oposes, algebr aic geometry and logic (Conf., Dalhou sie Univ ., Halifax, N. S., 1971) , pages 97–136. Lecture Notes in Math., V ol. 274. Springer , Berlin, 1972. [W ag94] Kim Ritter W agner . Solving Recursive Doma in Equations wi th Enrich ed Categ ories . PhD thesis, Carnegie Mellon Univ ersity , 1994. [W as02] Pawel W aszkiewicz. Quantitative C ontinuous Domains . PhD thesis, School of Computer S cience, The Univ ersity of Birmingham, Edgbaston, Birmingham B15 2TT , May 2002. [W oo04] R.J. W ood. Ordered sets via adjunctions. In Cate gorica l f oundations , volume 97 of Encyclope dia Math. Appl. , pages 5–47. Cambridge Univ . P ress, Cambridge, 2004. D ep ar t amento de M a tem ´ atica , U niversidad e de A veiro , 3810-193 A veiro , P or tugal E-mail addr ess : dirk@ua.pt