On the categorical meaning of Hausdorff and Gromov distances, I

ON THE CA TEGORICAL MEANING OF HA USDORFF AND GR OMOV DIST ANCES, I ANDREI AKHVLEDIANI, MARIA M ANUEL CLEMENTINO, AND W AL TER THOLEN Abstract. Hausdorff and Gromo v distances are in troduced and treated in the con text of c ategories enric hed o v er a c ommu tativ e unital quan tale V . The Hausdorff f unctor whic h, for ev ery V -cate gory X , provides the p ow erset of X with a suitable V -cat egory structure, is part of a monad on V - Cat whose Eilenberg-Mo ore algebras a re order-complete. The Gromov construction may be pursued for any endofuncto r K of V - Cat . In order to define the Gromo v “distance” betw een V -categories X and Y w e use V -modules betw een X and Y , rather than V -catego ry structures on the disjoi n t union of X and Y . Hence, we first pro vide a general exten sion theorem whic h, for an y K , yields a lax extension ˜ K to th e cate gory V - Mod of V - categories, wi th V -mo dules as mor- phisms. 1. Introduction The Hausdorff metric for (closed) s ubsets o f a (compact) metric space has b een recognized f or a long time as an imp or tant concept in many branches of math- ematics, and its origins rea ch back even beyond Hausdorff [9], to Pompeiu [13]; for a moder n account, see [2]. It has gained re newed int erest thro ugh Gromov’s work [8]. The Gr o mov-Hausdorff distance of tw o (compact) metric spaces is the infim um of their Hausdorff distances after having been isometrically embedded into any common lar ger s pace. There is therefore a notion of conv ergence for (isometry classes of compac t) metric spaces whic h has not only b ecome an impo rtant to ol in analysis and g eometry , but w hich has also pr ovided the key instrument for the pro of of Gro mov’s exis tence theorem for a nilpotent subgr oup of finite index in every finitely- generated g roup of po lynomial growth [7]. By interpreting the (non-negative) distances d ( x, y ) as hom( x, y ) and, hence, by rewriting the conditions 0 ≥ d ( x, x ) , d ( x, y ) + d ( y , z ) ≥ d ( x, z ) ( ∗ ) as k → hom( x, x ) , hom( x, y ) ⊗ hom( y , z ) → hom( x, z ) , Lawv ere [12] describ ed metric s paces as categories enriched ov er the (small and “thin”) symmetric mono idal-closed category P + = (([0 , ∞ ] , ≥ ) , + , 0), and in the F or e w ord of the electronic “ reprint” of [12] he s uggested that the Ha us dorff and Gromov metr ics sho uld b e develop ed for an ar bitrary symmetric mono idal-closed Date : No v em ber 1, 2018. 2000 Mathematics Subje ct Classific ation. Primary 18E40; Secondary 18A99. The first author ackn o wledges partial financial assistance from NSERC. The second author ackno wledges financial support fr om the Center of Mathematics of the Unive rsity of Coimbra/F CT. The third author ackn o wledges partial financial assistance from NSERC. 1 2 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN category ( V , ⊗ , k ). In this pap er we pr esent no tions o f Hausdorff and Gro mov dis- tance for the ca s e that V is “ thin”. Hence, w e replace P + by a commutativ e and unital quantale V , that is: by a complete lattice which is also a co mm utative monoid ( V , ⊗ , k ) such that the binar y op era tion ⊗ pr eserves suprema in each v ar ia ble. Put differently , we try to give answers to questions of the type: which structur e and prop erties of the (extended) non-negative r eal half-line allow for a meaningful tr eat- men t of Hausdo rff a nd Gromov distances, and which are their appropriate ca rrier sets? W e find that the guidance provided by enriched ca tegory theor y [11] is almost indispe nsable for finding satisfactory answers, and that so-c a lled ( bi- ) mo dules (or distributors ) betw een V -catego ries provide an elegant to ol for the theor y whic h may easily b e ov erlo oked without the c a tegorica l environmen t. Hence, our prima r y mo- tiv ation for this work is the desire fo r a b etter understanding of the tr ue ess ent ials of the class ical metric theory and its applications, r ather than the desire for giving merely a more genera l framework which, how ev er, may prov e to b e useful as well. Since ( ∗ ) iso lates pr ecisely those conditions of a metric whic h lend themselves naturally to the hom interpretation, a discussio n of the others seems to b e nec essary at this point; these ar e: − d ( x, y ) = d ( y , x ) ( symmetry ), − x = y whenever d ( x, y ) = 0 = d ( y , x ) ( sep ar ation ), − d ( x, y ) < ∞ ( finiteness ). With the distance of a p oint x to a subset B o f the metric spa ce X = ( X , d ) b e given by d ( x, B ) = inf y ∈ B d ( x, y ), the non-symmet ric Hausdorff distanc e from a subset A to B is defined by H d ( A, B ) = sup x ∈ A d ( x, B ) , from which o ne obtains the class ical Hausdorff distanc e H s d ( A, B ) = max { H d ( A, B ) , H d ( B , A ) } by enfor c e d symmetrization. But not only symmetry , but also separation and finite- ness get los t under the ra ther natura l passa ge from d to H d . (If one thinks of d ( x, B ) as the travel time from x to B , then H d ( A, B ) may b e though t of as the time needed to ev a cuate everyone living in the area A to the are a B .) In order to sav e them one usually res tricts the carrier set fro m the entire p ow erset P X to the closed subsets of X (whic h makes H s d sepa rated), or even to the non-empt y compact subsets (whic h guara nt ees a lso finiteness). As in [10] we ca ll a P + -categor y an L -metric sp ac e , that is a set X equipp ed with a function d : X × X → [0 , ∞ ] sa tisfying ( ∗ ); a P + -functor f : ( X , d ) → ( X ′ , d ′ ) is a non-expa nsive map, e .g . a ma p f : X → X ′ satisfying d ′ ( f ( x ) , f ( y )) ≤ d ( x, y ) for a ll x, y ∈ X . That the underlying- s et functor makes the resulting categor y Met t op olo gic al ov er Set (see [5]) provides furthe! r evidence that prop e rties ( ∗ ) are fundamental and are b etter c o nsidered sepa rately from the other s, even thoug h symmetry (as a coreflective pro pe rty) would not ob- struct to po logicity . But inclusion o f (the reflective pro pe rty o f ) separation would, and inclusion of (the neither r e flective nor coreflective pr op erty of ) finiteness would make for an ev en p o ore r catego rical environment. While symmetry see ms to b e ar tificially sup erimp osed on the Hausdo rff metric, it plays a crucial r ole for the Gr omov distance, which b ecomes evident alr eady when we lo ok a t the mos t elementary examples . Initially nothing pr e ven ts us from ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 3 considering arbitrar y L -metric spaces X , Y a nd putting GH ( X , Y ) = inf Z H d Z ( X, Y ) , where Z runs thro ugh all L -metric spa ces Z into which b oth X and Y are is omet- rically embedded. But for X = { p } a single ton se t and Y = { x, y , z } three equidis- tant points, with all distance s 1, say , for every ε > 0 we can mak e Z = X ⊔ Y a (prop er ) metric spa ce, with d ( p, x ) = d ( x, p ) = ε and all other non-zero dis- tances 1. Then H d Z ( X, Y ) = ε , and GH ( X , Y ) = 0 follows. One has also GH ( Y , X ) = 0 but here one needs non-symmetric (but still separa ted) structures: put d ( x, p ) = d ( y , p ) = d ( z , p ) = ε , but let the reverse distances b e 1 . Hence, even a p osteriori symmetriza tio n lea ds to a trivial distance betw een non-isomorphic spa c e s. How e ver, there are tw o wa ys of a priori symmetrization w hich b oth yield the in- tuitiv ely desir ed result 1 2 for the Gromov distance in this ex ample: One w a y is b y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y z p 1 1 1 1 2 1 2 1 2 restricting the rang e of the infimum in the definition of GH ( X, Y ) to symmetric L -metric spaces, which seems to be na tural when X and Y are symmetric. (Indeed, if for our exa mple spaces one as sumes H d Z ( Y , X ) < 1 2 with d Z symmetric, then the tria ngle inequality would b e v io lated: 1 ≤ d ( x, p ) + d ( p, y ) < 1 2 + 1 2 .) The other wa y “ works” a ls o for non-s y mmetric X and Y ; one simply puts GH s ( X, Y ) = inf Z H s d Z ( X, Y ) , with Z running as in GH ( X , Y ). (When H d Z ( Y , X ) ≤ 1 2 , then 1 2 = 1 − 1 2 ≤ min { d ( p, x ) , d ( p, y ) , d ( p, z ) } = H d Z ( X, Y ) ≤ H s d Z ( X, Y ) , and when H d Z ( Y , X ) ≥ 1 2 , then tr iv ially 1 2 ≤ H s d Z ( X, Y ).) Having r e cognized H (and H s ) as endofunctor s of Met , these consideratio ns suggest that G is a n “op era tor” on such endofunctors. But in or der to “compute” its v alues, one needs to “control” the spaces Z in their defining formu la, a nd here is where modules come in. (A module be t w een L -metric spaces generalizes a non- expansive map just lik e a relation generalizes a mapping b etw een sets.) A mo dule from X to Y cor resp onds to a n L -metric that one may impos e o n the disjoin t unio n X ⊔ Y . T o take adv an tage o f this viewp oint, it is neces sary to extend H from non- expansive maps to mo dules (leaving its o per ation on ob jects unchanged) to be come a la x functor ˜ H . Hence, GH ( X, Y ) may then b e more compactly defined using an infim um that ranges just ov er the hom-s et of mo dules fro m X to Y . In Section 2 we give a br ie f ov erview o f the needed to ols from enriched catego ry theory , in the highly simplified context of a qua nt ale V . The purp ose of Section 3 is to establis h a g eneral extension theor em for endofunctors of V - Cat , so that 4 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN they can act on V -mo dules rather than just o n V -functors. In Sectio ns 4 and 5 we consider the Hausdo rff mo na d of V - Cat and its lax extensio n to V -mo dules, and we determine the Eilen ber g-Mo ore ca teg ory in b oth cases. The Gromov “distanc e ” is considered for a fairly genera l r ange of endofunctors of V - Cat in Sec tion 6, and the resulting Gro mov “spa ce” of is o morphism cla s ses of V -categ ories is presented a s a large colimit. F or the endofunctor H , in Section 7 this larg e “space” is shown to carry in ternal monoid structures in the monoidal category V - CA T which allow us to consider H as an in ternal homomorphism. The effects of symmetrization and the status of separatio n are discussed in Sections 8 and 9. The fundamental question of transfer of (Cauch y-)completeness from X to H X , as well as the question of completeness of suitable subspaces o f the Gromov “ space” will b e considered in the second part of this pap er. The r e a der is r eminded that, since P + carries the natural ≥ as its o r der, in the context o f a g e neral quantale V the na tur al infima and suprema of P + app ear a s joins ( W ) and meets ( V ) in V . While this ma y a ppea r to be irr itating initially , it reflects in fact the logical vie w p o int dictated by the ele mentary case V = 2 = ( {⊥ < ⊤} , ∧ , ⊤ ), and it translates bac k w ell even in the metric case. (F or example, if we write the sup-metric d of the re a l function space C ( X ) as d ( f , g ) = ^ x ∈ X | f ( x ) − g ( x ) | , then the statement d ( f , g ) = 0 ⇐ ⇒ for al l x ∈ X : f ( x ) = g ( x ) seems to r ead off the defining formula mor e dire ctly .) A cknow le dgments While the work presented in this pa per fir st b ega n to take shap e when, aimed with her knowledge of the treatment of the Hausdo r ff metric in [3], the second-named author visited the third in the Spr ing o f 200 8, w hich then gav e rise to a muc h mor e comprehensive study b y the first-named author in his Master’s thesis [1] that co nt ains many elemen ts of the curre n t work, precurso rs of it go in fact ba ck to a vis it by Richard W o o d to the third-named author in 200 1. How ev er, the attempt to work immediately with a (non-thin) symmetric mo noidal-close d category proved to be to o difficult at the time. The seco nd- and third-named authors also acknowledge encoura gement and fer tile p ointers given by Bill Lawvere ov er the years, es p ecia lly after a talk of the third-na med autho r at the Roy al Flemish Academy of Sciences in Oc to ber 2008 . This ta lk also led to a most interesting exchange with Isa r Stubbe who meanwhile has ca rried the theme of this pa pe r int o the more gener a l context w her eby the quantale V is tr aded fo r a quantaloid Q (see [14]), a clea r indication that the ca tegorica l study of the Hausdorff a nd Gromov metric may still b e in its infancy . 2. Quant ale-enriched ca tegories Throughout this pap er, V is a commutativ e, unita l quantale. Hence, V is a complete lattice with a c ommut ative, asso ciative binary op e r ation ⊗ and a ⊗ - neutral elemen t k , such tha t ⊗ preser ves arbitrar y suprema in each v ariable. Our paradigma tic examples 2 =  {⊥ < ⊤} , ∧ , ⊤  and P + =  ([0 , ∞ ] , ≥ ) , + , 0  were a lr eady considered by Lawv ere [12]; they serve to provide bo th an o rder- theoretic and a metric intuition for the theo ry . ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 5 A V -r elation r from a set X to a set Y , written as r : X − → 7 Y , is simply a function r : X × Y → V . Its co mpo sition with s : Y − → 7 Z is g iven by  s · r  ( x, z ) = _ y ∈ Y r ( x, y ) ⊗ s ( y , z ) . This defines a ca tegory V - Rel , and there is an obvious functor Set → V - Rel which assigns to a ma pping f : X → Y its V -graph f ◦ : X − → 7 Y with f ◦ ( x, y ) = k if f ( x ) = y , and f ◦ ( x, y ) = ⊥ o ther wise. This functor is faithful o nly if k > ⊥ , which we will assume hence fo rth, wr iting just f for f ◦ . There is an in volution ( ) ◦ : V - Rel op → V - Rel whic h sends r : X − → 7 Y to r ◦ : Y − → 7 X with r ◦ ( y , x ) = r ( x, y ). With the p oint wise order of its hom-sets, V - Rel becomes o rder-enr iched, e.g. a 2-catego ry , and mappings f : X → Y b ecome maps in the 2 -categor ical sense: 1 X ≤ f ◦ · f , f · f ◦ ≤ 1 Y . A V -c ate gory X = ( X , a ) is a set X with a V -relation a : X − → 7 X satisfying 1 X ≤ a a nd a · a ≤ a ; element wis e this mea ns k ≤ a ( x, x ) and a ( x, y ) ⊗ a ( y , z ) ≤ a ( x, z ) . A V -functor f : ( X, a ) → ( Y , b ) is a map f : X → Y with f · a ≤ b · f , or eq uiv alen tly a ( x, y ) ≤ b  f ( x ) , f ( y )  for all x, y ∈ X . The re sulting category V - Cat yields the categ ory Ord of (pre)ordere d sets and monoto ne ma ps for V = 2 and the category Met of L -metric spaces for V = P + . V - Cat has a symmetric monoidal-closed structur e, given by ( X, a ) ⊗ ( Y , b ) = ( X × Y , a ⊗ b ) , X – ◦ Y =  V - Cat ( X , Y ) , c  with a ⊗ b  ( x, y ) , ( x ′ , y ′ )  = a ( x, x ′ ) ⊗ b ( y , y ′ ) , c ( f , g ) = ^ x ∈ X b  f ( x ) , g ( x )  . Note that X ⊗ Y must b e distinguished from the Car tesian pro duct X × Y whos e structure is a × b with a × b  ( x, y ) , ( x ′ , y ′ )  = a ( x, x ′ ) ∧ b ( y , y ′ ) . V itself is a V - category with its “in ternal hom” – ◦ , given by z ≤ x – ◦ y ⇐ ⇒ z ⊗ x ≤ y for all x, y , z ∈ V . The mo rphism 2 → V of quantales has a r ight a djo int V → 2 that sends v ∈ V to ⊤ precisely when v ≥ k . Hence, there is an induced functor V - Cat → Ord which pr ovides a V -catego ry with the o rder x ≤ y ⇐ ⇒ k ≤ a ( x, y ) . Since V - R el ( X, Y ) = V X × Y = ( X × Y ) – ◦ V 6 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN is a V -ca tegory (as a pro duct of ( X × Y )-many copies of V , o r as a “function space” with X , Y discrete), it is easy to show that V - Rel is  V - Cat  -enriched, e.g. E → V - Rel ( X , X ) , V - Rel ( X , Y ) ⊗ V - Rel ( Y , Z ) → V - Rel ( X , Z ) ∗ 7→ 1 X , ( r , s ) 7→ s · r are V -functors (where E = ( {∗ } , k ) is the ⊗ - unit in V - Cat ). 3. Modules, Extensio n Theorem F or V -catego ries X = ( X , a ) , Y = ( Y , b ) a V -(bi)mo du le (a ls o: V - distributor , V - pr ofunctor ) ϕ from X to Y , written as ϕ : X − → ◦ Y , is a V -relation ϕ : X − → 7 Y with ϕ · a ≤ ϕ a nd b · ϕ ≤ ϕ , that is a ( x ′ , x ) ⊗ ϕ ( x, y ) ≤ ϕ ( x ′ , y ) and ϕ ( x, y ) ⊗ b ( y , y ′ ) ≤ ϕ ( x, y ′ ) for a ll x, x ′ ∈ X , y , y ′ ∈ Y . F or ϕ : X − → ◦ Y one actually has ϕ · a = ϕ = b · ϕ , so that 1 ∗ X := a plays the role of the identit y mor phis m in the ca teg ory V - Mo d of V -ca tegories (as ob jects) and V -mo dules (as morphisms). It is easy to show that a V -r elation ϕ : X − → ◦ Y is a V -mo dule if, and only if, ϕ : X op ⊗ Y → V is a V -functor (see [4 ]); here X op = ( X , a ◦ ) for X = ( X , a ). Hence, V - Mo d ( X , Y ) =  X op ⊗ Y  – ◦ V . In par ticular V - Mo d is (like V - Rel ) V - Cat - enriche d . Also, V - Mo d inherits the 2 - categoric al s tructure from V - R el , just v ia p oint wise o rder. Every V -functor f : X → Y induces adjoint V -mo dules f ∗ ⊣ f ∗ : Y − → ◦ X with f ∗ := b · f , f ∗ := f ◦ · b (in V - Rel ). Hence, ther e a r e functor s ( − ) ∗ : V - Cat → V - Mod , ( − ) ∗ : V - Cat op → V - Mo d which map ob jects identically . V - Cat b ecomes order-enriched (a 2-ca tegory) via f ≤ g ⇐ ⇒ f ∗ ≤ g ∗ ⇐ ⇒ ∀ x : f ( x ) ≤ g ( x ) . The V -functor f : X → Y is ful ly faithful if f ∗ · f ∗ = 1 ∗ X ; equiv a len tly , if a ( x, x ′ ) = b  f ( x ) , f ( x ′ )  for all x, x ′ ∈ X . Via ϕ : X − → ◦ Y X op ⊗ Y → V y ϕ : Y → ( X op – ◦ V ) =: ˆ X , every V -mo dule ϕ corr esp onds to its Y one da mate y ϕ in V - Cat . In par ticular, a = 1 ∗ X corres p o nds to the Y oneda functor y X = y 1 ∗ X : X → ˆ X . F or every V -functor f : X op → V and x ∈ X o ne has 1 ∗ ˆ X ( y X ( x ) , f ) = f ( x ) (Y one da L emma) . In par ticular, 1 ∗ ˆ X  y X ( x ) , y X ( x ′ )  = a ( x, x ′ ), i.e. y X is fully faithful. The corr esp ondence b etw een ϕ and y ϕ gives: ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 7 Prop ositio n 1. ( − ) ∗ :  V - Cat  op → V - Mo d has a left adjoint ˆ ( − ) , give n by ˆ ϕ ( s )( x ) = _ y ∈ Y ϕ ( x, y ) ⊗ s ( y ) for al l ϕ : X − → ◦ Y , s ∈ ˆ Y , x ∈ X . Pr o of. Under the c o rresp ondence ϕ : X − → ◦ Y Φ : Y → ˆ X given by ϕ ( x, y ) = Φ( y )( x ), Φ = 1 ˆ X gives the unit η X : X − → ◦ ˆ X of the a djunction, with η X ( x, t ) = t ( x ) for all x ∈ X , t ∈ ˆ X . Note that o ne has η X = ( y X ) ∗ , by the Y o neda Lemma. W e m ust co nfir m that y ϕ is indeed the unique V -functor Φ : Y → ˆ X with Φ ∗ · η X = ϕ . But any such Φ must satisfy ϕ ( x, y ) =  Φ ∗ · ( y X ) ∗  ( x, y ) = _ t ∈ ˆ X 1 ∗ ˆ X  y X ( x ) , t ) ⊗ 1 ∗ ˆ X  t, Φ( y )  ≤ _ t ∈ ˆ X 1 ∗ ˆ X  y X ( x ) , Φ( y )  ≤ Φ( y )( x ) ≤ 1 ∗ ˆ X  y X ( x ) , y X ( x )  ⊗ 1 ∗ ˆ X  y X ( x ) , Φ( y )  ≤  Φ ∗ · ( y X ) ∗  ( x, y ) = ϕ ( x, y ) for all x ∈ X, y ∈ Y . Hence, necessar ily Φ = y ϕ , and the same calculation shows ϕ = ( y ϕ ) ∗ · η X . Now, ˆ ϕ : ˆ Y → ˆ X is the V -functor corres po nding to η Y · ϕ , hence ˆ ϕ ( s )( x ) = ( η Y · ϕ )( x, s ) = _ y ∈ Y ϕ ( x, y ) ⊗ s ( y ) , for all s ∈ ˆ Y , x ∈ X .  R emarks 1 . (1) F or ϕ : X − → ◦ Y , the V -functor ˆ ϕ may a lso b e describ ed a s the left Kan extensio n of y ϕ : Y → ˆ X along y Y : Y → ˆ Y . (2) T he adjunction of Pro po sition 1 is in fa c t 2- categorica l. It therefore induces a 2-mo na d P V = ( P V , y , m ) on V - Cat , with P V X = ˆ X = X – ◦ V , P V f = c f ∗ : ˆ X → ˆ Y for f : X → Y = ( Y , b ), where c f ∗ ( t )( y ) = _ x ∈ X b  y , f ( x )  ⊗ t ( x ) for t ∈ ˆ X , y ∈ Y . This monad is of Ko ck-Z¨ ob erlein type, i.e. one has c y ∗ X ≤ y ˆ X : ˆ X → ˆ ˆ X . 8 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN In fact, for all x, y ∈ X = ( X , a ), and t, s ∈ ˆ X one ha s a ( x, y ) ≤ s ( x ) – ◦ s ( y ), hence t ( y ) ⊗  t ( y ) – ◦ a ( x, y )  ⊗ s ( x ) ≤ a ( x, y ) ⊗ s ( x ) ≤ s ( y ) , which gives c y ∗ X ( s )( t ) = _ x y ∗ X ( t, x ) ⊗ s ( x ) = _ x ^ y  t ( y ) – ◦ a ( x, y )  ⊗ s ( x ) ≤ ^ y t ( y ) – ◦ s ( y ) = y ˆ X ( s )( t ) . (3) T he adjunction of Pro p o sition 1 induces also a monad o n V - Mo d which we will not consider further in this pap er . But see Section 5 b elow. (4) B ecause of (2), the Eilenber g -Mo ore categ ory ( V - Cat ) P V has V -categ ories X as ob jects which come equippe d with a V -functor α : ˆ X → X w ith α · y X = 1 X and 1 ˆ X ≤ y X · α , e.g V -ca tegories X for which y X has a left adjoint. These are known to b e the V -catego ries that have all weighte d c oli mits (see [11]), with α pr oviding a c hoice o f such colimits. Morphisms in ( V - Cat ) P V m ust preserve the (chosen) weigh ted colimits. (5) I n case V = 2 , P V X ca n be iden tified with the set P ↓ X o f down-closed subsets of the (pre)o rdered set X , and the Y oneda functor X → P ↓ X sends x to its down-closure ↓ x . Note that P ↓ X is the ordina r y p ow er set P X of X when X is discr ete. Ord P ↓ has complete ordered s ets as ob jects, and its mor phis ms must preserve supr ema. Hence, this is the categor y Sup of so-called sup-lattic es (with no anti-symmetry condition). Next w e pr ove a general extens ion theo rem fo r endofunctors of V - Cat . While maintaining its effect on o b jects, we wish extend any functor K defined for V - functors to V -mo dules . T o this end w e observe that for a V -mo dule ϕ : X − → ◦ Y , the left triangle of ˆ Y ˆ ϕ   > > > > > > > ˆ Y ˆ ϕ   > > > > > > > Y y Y @ @         y ϕ / / ˆ X Z y ψ @ @         y ψ · ϕ / / ˆ X commutes, since y Y is the counit of the adjunction o f Pro p os ition 1. More generally , the right tria ngle commutes for every ψ : Y − → ◦ Z . Theorem 1 (Extension Theorem) . F or every functor K : V - Cat → V - Cat , ˜ K ϕ :=  K X ◦ ( K y X ) ∗ / / K ˆ X ◦ ( K y ϕ ) ∗ / / K Y  ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 9 defines a lax functor ˜ K : V - Mo d → V - Mo d which c oincides with K on obje cts. Mor e over, if K pr eserves ful l fidelity of V -functors, the diagr am V - Mo d ˜ K / / V - Mo d ( V - Cat ) op ( − ) ∗ O O K op / / ( V - Cat ) op ( − ) ∗ O O c ommut es. Pr o of. L a x functor iality of ˜ K follows from ˜ K (1 ∗ X ) = ( K y X ) ∗ · ( K y X ) ∗ ≥ 1 ∗ K X , ˜ K ( ψ · ϕ ) = ( K y ψ · ϕ ) ∗ · ( K y X ) ∗ = ( K y ψ ) ∗ · ( K ˆ ϕ ) ∗ · ( K y X ) ∗ ≥ ( K y ψ ) ∗ · ( K y Y ) ∗ · ( K y ϕ ) ∗ · ( K y X ) ∗ = ˜ K ψ · ˜ K ϕ, since y ϕ = ˆ ϕ · y Y implies ( K y ϕ ) ∗ = ( K y Y ) ∗ · ( K ˆ ϕ ) ∗ , hence ( K ˆ ϕ ) ∗ ≥ ( K y Y ) ∗ · ( K y ϕ ) ∗ by a djunction. F or a V -functor f : X → Y , the triangle Y y Y @ @ @ @ @ @ @ X f ? ?         y f ∗ / / ˆ Y . commutes, so that ˜ K ( f ∗ ) = ( K y f ∗ ) ∗ · ( K y Y ) ∗ = ( K f ) ∗ ( K y Y ) ∗ ( K y Y ) ∗ ≥ ( K f ) ∗ , and o ne even has ˜ K ( f ∗ ) = ( K f ) ∗ if K pres erves the full fidelity o f y Y .  4. The Hausdorff Monad on V - Cat Let X = ( X , a ) be a V -catego ry . Then ˆ X = ( X op – ◦ V ) = P V X is closed under suprema formed in the pro duct V X ; hence, like V it is a sup-lattice. Consequently , the Y oneda functor y X : X → ˆ X factors uniquely through the free sup-lattice P X , by a sup- preserving map Y X : P X → P V X : X {−} / / y X ! ! D D D D D D D D P X Y X   B _   P V X a ( − , B ) where a ( x, B ) = _ y ∈ B a ( x, y ) for all x ∈ X , B ⊆ X . W e can provide the se t P X with a V -categor y structure h X which it inher its from P V X (since the fo r getful functor V - Cat → Set is a fibration, 10 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN even a to po logical functor, s ee [5]). Hence, for subsets A, B ⊆ X one puts h X ( A, B ) = ^ z ∈ X a ( z , A ) – ◦ a ( z , B ) . Lemma 1. h X ( A, B ) = ^ x ∈ A _ y ∈ B a ( x, y ) . Pr o of. F rom k ≤ a ( x, A ) for all x ∈ A one o btains h X ( A, B ) ≤ ^ x ∈ A a ( x, A ) – ◦ a ( x, B ) ≤ ^ x ∈ A k – ◦ a ( x, B ) = ^ x ∈ A a ( x, B ) . Conv ersely , with v := ^ x ∈ A _ y ∈ B a ( x, y ) , we m ust show v ≤ a ( z , A ) – ◦ a ( z , B ) for a ll z ∈ X . But since for every x ∈ A a ( z , x ) ⊗ v ≤ a ( z , x ) ⊗ _ y ∈ B a ( x, y ) = _ y ∈ B a ( z , x ) ⊗ a ( x, y ) ≤ a ( z , B ) , one concludes a ( z , A ) ⊗ v ≤ a ( z , B ), as desire d.  F or a V -functor f : X → Y = ( Y , b ) one no w concludes ea sily h X ( A, B ) ≤ ^ x ∈ A _ y ∈ B b  f ( x ) , f ( y )  = h Y  f ( A ) , f ( B )  for all A, B ⊆ X . Consequently , with H X = ( P X, h X ) , H f : H X → H Y , A 7→ f ( A ) , one obtains a (2-)functor H whic h makes the dia gram V - Cat H / /   V - Cat   Set P / / Set commute. Actually , one has the following theor em: Theorem 2. Th e p owe rset monad P = ( P, {−} , S ) c an b e lifte d along the for get fu l functor V - Cat → Set t o a monad H of V - Cat of Ko ck-Z¨ ob erlein typ e. Pr o of. F or a V -c ategory X , x 7→ { x } gives a fully faithful V -functor {−} : X → H X . In or der to s how tha t [ : H H X → H X , A 7→ [ A , is a V -functor, it suffices to verify that for all x ∈ A ∈ A ∈ H H X and B ∈ H H X one has h H X ( A , B ) ≤ a ( x, [ B ) . But for all B ∈ B we have h X ( A, B ) ≤ a ( x, B ) ≤ a ( x, [ B ) , so that h H X ( A , B ) ≤ _ B ∈B h X ( A, B ) ≤ a ( x, [ B ) . ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 11 The induced o rder of H X is given by A ≤ B ⇐ ⇒ ∀ x ∈ A : k ≤ a ( x, B ) , and that of H H X by A ≤ B ⇐ ⇒ ∀ A ∈ A : k ≤ _ B ∈B h X ( A, B ) . Hence, fro m k ≤ a ( x, A ) = h X ( { x } , A ) for a ll A ∈ H X one obtains  { x } | x ∈ A  ≤ { A } in H H X , which means H { −} X ≤ {−} H X , i.e., H is K o ck-Z¨ ob erlein.  R emarks 2 . (1) B y definition, h X ( A, B ) dep ends only o n a ( − , A ), a ( − , B ). Hence, if we put ⇓ X B :=  x ∈ X | { x } ≤ B  = { x ∈ X | ↓ x ≤ B } = { x ∈ X | k ≤ a ( x, B ) } , from B ⊆⇓ X B one trivially ha s a ( z , B ) ≤ a ( z , ⇓ X B ) for all z ∈ X , but also a ( z , ⇓ X B ) = _ x ∈⇓ X B a ( z , x ) ⊗ k ≤ _ z ∈⇓ X B _ y ∈ B a ( z , x ) ⊗ a ( x, y ) ≤ a ( z , B ) . Consequently , h X ( A, B ) = h X ( ⇓ X A, ⇓ X B ) . This equation also implies ⇓ X ⇓ X B = ⇓ X B . (2) ⇓ X B of (1) m ust not be confused with the down-closure ↓ X B of B in X w.r.t the induced or der of X , e.g. with ↓ X B = { x ∈ X | ∃ y ∈ B x ≤ y } = { x ∈ X | ∃ y ∈ B ( k ≤ a ( x, y )) } . In genera l, B ⊆ ↓ X B ⊆⇓ X B . While ↓ X B = ⇓ X B for V = 2 , the t w o se ts are genera lly distinct even for V = P + . (3) I n the induced o rder o f H X o ne has A ≤ B ⇐ ⇒ A ⊆⇓ X B . Hence, if we restrict H X to H ⇓ X := { B ⊆ X | B = ⇓ X B } , the induced order of H ⇓ X is simply the inclusio n order . H ⇓ bec omes a functor H ⇓ : V - Cat → V - Cat with ( H ⇓ f )( A ) = ⇓ Y f ( A ) for all A ∈ H ⇓ X , and there is a lax natural tra nsformation ι : H ⇓ → H given b y inclusion functions. Like H , also H ⇓ carries a monad structure, given by X → H ⇓ X , x 7→↓ X x = ⇓ X x, H ⇓ H ⇓ X → H ⇓ X , B 7→⇓ X ( [ B ) . In this wa y ι : H ⇓ → H b ecomes a lax mona d mor phism. 12 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN (4) B y definitio n, y X is fully faithful. Hence, H X car ries the larges t V -categ ory structure making y X : H X → P V X a V -functor. Equiv a lently , this is the largest V -categor y structure making δ X : X − → ◦ H X with δ ( x, B ) = a ( x, B ) a V -mo dule. (5) Y X : H X → P V X defines a mor phism H → P V of mona ds. Indee d, the left diagram of X {−} } } { { { { { { { { y X ! ! D D D D D D D D H H X S   H Y X / / H P V X Y ˆ X / / P V P V X m X   H X Y X / / P V X H X Y X / / P V X commutes tr ivially , a nd for the right one first o bs erves that m X : ˆ ˆ X → ˆ X is defined by m X ( τ )( x ) = c η X ( τ )( x ) = _ t ∈ ˆ X t ( x ) ⊗ τ ( t ) for all τ ∈ ˆ ˆ X , x ∈ X . Hence, for B ∈ H H X we hav e: ( m X · Y ˆ X · H Y X )( B )( x ) = _ t ∈ ˆ X t ( x ) ⊗ Y ˆ X  Y X ( B )  = _ t ∈ ˆ X t ( x ) ⊗ 1 ∗ ˆ X  t, Y X ( B )  = _ t ∈ ˆ X _ B ∈B t ( x ) ⊗  ^ x ′ ∈ X t ( x ′ ) – ◦ a ( x ′ , B )  ≤ _ B ∈B _ t ∈ ˆ X t ( x ) ⊗  t ( x ) – ◦ a ( x, B )  = _ B ∈B a ( x, B ) = Y X ( [ B )( x ) ≤ a ( x, x ) ⊗ _ B ∈B 1 ∗ ˆ X  y X ( x ) , a ( − , B )  ≤ _ t ∈ ˆ X t ( x ) ⊗ 1 ∗ ˆ X  t, Y X ( B )  = ( m X · Y ˆ X · H Y X )( B )( x ) . Consequently , ther e is an induced algebr a ic functor ( V - Cat ) H → ( V - Cat ) P V of the resp ective Eile nber g-Mo or e ca tegories. W e brie fly descr ib e the E ilenberg-Mo or e ca tegory ( V - Cat ) H ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 13 where ob jects X ∈ V - Cat come equipp ed with a V -functor α : H X → X satisfying α · {− } = 1 X and 1 H X ≤ {−} · α (since H is Ko ck-Z¨ o be r lein). Hence, α ( { x } ) = x for all x ∈ X , and A ≤ { α ( A ) } for A ∈ H X , that is: k ≤ h X ( A, { α ( A ) } ) = ^ x ∈ A a  x, α ( A )  . Consequently , α ( A ) is an upp er b ound of A in the induced o rder o f X , and for any other upp e r b o und y of A in X = ( X , a ) o ne has k ≤ ^ x ∈ A a ( x, y ) = h X ( A, { y } ) ≤ a  α ( A ) , α ( { y } )  = a  α ( A ) , y  since α is a V -functor . Hence, α ( A ) gives (a choice o f ) a supremum of A in X . Moreov er, the la s t computation shows ( ∗ ) a ( _ A, y ) = ^ x ∈ A a ( x, y ) for all y ∈ X , A ∈ H X (since “ ≤ ” holds trivially ). Conversely , any V -categ o ry X = ( X , a ) which is complete in its induced order and satisfies ( ∗ ) is e asily seen to be an ob ject o f ( V - Cat ) H . Corollary 1. The Eilenb er g-Mo or e c ate gory of H has or der-c omplete V -c ate gories X = ( X , a ) satisfying ( ∗ ) as its obje cts, and morphisms ar e V -fun ctors pr eserving (the chosen) supr ema. 5. The lax Hausdorff monad on V - Mo d When a pplying Theorem 1 to the Hausdo rff functor H : V - Cat → V - Cat of Theorem 2 we obtain a lax functor ˜ H : V - Mo d → V - Mo d whose v alue on a V - mo dule ϕ : X − → ◦ Y may b e ea sily computed: Lemma 2. ˜ H ϕ ( A, B ) = ^ x ∈ A _ y ∈ B ϕ ( x, y ) for al l subsets A ⊆ X , B ⊆ Y . 14 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN Pr o of. ˜ H ϕ ( A, B ) = _ D ∈ H ˆ X ( H y X ) ∗ ( A, D ) ⊗ ( H y ϕ ) ∗ ( D , B ) = _ D ∈ H ˆ X h ˆ X  y X ( A ) , D  ⊗ h ˆ X  D , y ϕ ( B )  ≤ h ˆ X  y X ( A ) , y ϕ ( B )  = ^ x ∈ A _ y ∈ B 1 ∗ ˆ X  y X ( x ) , y ϕ ( y )  = ^ x ∈ A _ y ∈ B ϕ ( x, y ) ( Y oneda ) = h ˆ X  y X ( A ) , y ϕ ( B )  ≤ h ˆ X  y X ( A ) , y X ( A )  ⊗ h ˆ X  y X ( A ) , y ϕ ( B )  ≤ _ D ∈ H ˆ X h ˆ X  y X ( A ) , D  ⊗ h ˆ X  D , y ϕ ( B )  = ˜ H ϕ ( A, B ) .  W e now prove that ˜ H carr ies a lax monad structure. Theorem 3. ˜ H b elongs to a lax monad ˜ H = ( ˜ H , δ, ν ) of V - Mo d such that H of The or em 2 is a lifting of ˜ H along ( − ) ∗ : V - Cat → V - Mod . Pr o of. L e t us fir st note that H is a lifting of ˜ H along ( − ) ∗ , in the sense that V - Cat H / / ( − ) ∗   V - Cat ( − ) ∗   V - Mo d ˜ H / / V - Mo d commutes. Indeed, for f : X → Y = ( Y , b ) in V - Cat and A ∈ H X , B ∈ H Y one has ˜ H ( f ∗ )( A, B ) = ^ x ∈ A _ y ∈ B b  f ( x ) , y  = h Y  f ( A ) , B )  = ( H f ) ∗ ( A, B ) . The unit of ˜ H , δ : 1 → ˜ H , is defined by δ X = {−} ∗ : X − → ◦ H X , δ X ( x, B ) = h X ( { x } , B ) = a ( x, B ) , for X = ( X , a ), x ∈ X , B ∈ H X (see also Rema rks 2 (2)), and the multiplication ν : ˜ H ˜ H → ˜ H can b e given by ν X = [ ∗ : H H X − → ◦ H X , ν X ( A , B ) = h X ( [ A , B ) = ^ A ∈A h X ( A, B ) , for A ∈ H H X , B ∈ H X . The monad conditions ho ld strictly for ˜ H , b ecause they hold s tr ictly for H . F or exa mple, ν · ˜ H δ = 1 follows fro m ν X · ˜ H δ X = [ ∗ · ˜ H ( {−} ∗ ) = [ ∗ · ( H {−} ) ∗ = ( [ · H {− } ) ∗ = 1 ∗ X . ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 15 Surprisingly though, also the naturality sq uares for bo th δ X and ν X commute strictly . Indeed, for ϕ : X − → ◦ Y = ( Y , b ), x ∈ X , B ∈ H Y and A ∈ H H X one has: ( ˜ H ϕ · δ X )( x, B ) = _ A ∈ H X δ X ( x, A ) ⊗ ˜ H ϕ ( A, B ) = _ A ∈ H X h X ( { x } , A ) ⊗ ˜ H ϕ ( A, B ) = ˜ H ϕ ( { x } , B ) = _ y ∈ B ϕ ( x, y ) = _ y ∈ B _ z ∈ Y ϕ ( x, z ) ⊗ b ( z , y ) = _ z ∈ Y ϕ ( x, z ) ⊗  _ y ∈ B b ( z , y )  = _ z ∈ Y ϕ ( x, z ) ⊗ δ Y ( z , B ) = ( δ Y · ϕ )( x, B ) , ( ˜ H ϕ · ν X )( A , B ) = _ A ∈ H X ν X ( A , A ) ⊗ ˜ H ϕ ( A, B ) = _ A ∈ H X h X ( [ A , A ) ⊗ ˜ H ϕ ( A, B ) = ˜ H ϕ ( [ A , B ) ≤  ^ A ∈A _ B ′ ∈ H B ˜ H ϕ ( A, B ′ )  ⊗ ^ B ′ ∈ H B h Y ( B ′ , B ) (since k ≤ h Y ( B ′ , B ) fo r B ′ ∈ H B ) ≤ _ B∈ H H Y  ^ A ∈A _ B ′ ∈B ˜ H ϕ ( A, B ′ )  ⊗  ^ B ′ ∈B h Y ( B ′ , B )  = ( ν Y · ˜ H ˜ H ϕ )( A , B ) = _ B∈ H H Y ˜ H ˜ H ϕ ( A , B ) ⊗ ν Y ( B , B ) ≤ _ B∈ H H Y ^ A ∈A ˜ H ϕ ( A, [ B ) ⊗ h Y ( [ B , B ) ≤ ^ A ∈A ˜ H ϕ ( A, B ) = ( ˜ H ϕ · ν X )( A , B ) .  R emarks 3 . (1) W e emphasize that, while ˜ H is only a lax functor, this is in fact the only defect that pr even ts ˜ H from b eing a monad in the strict sense. (2) I n addition to the commu tativity of the diagram g iven in the P ro of o f Theorem 3, since H o bviously pr eserves full fidelity of V -functors, from 16 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN Theorem 1 we obtain als o the commutativit y of ( V - Cat ) op H op / / ( − ) ∗   ( V - Cat ) op ( − ) ∗   V - Mo d ˜ H / / V - Mo d (3) I f V is c onst r u ctively c ompletely distributive (see [15], [3]), then ˜ H ϕ for ϕ : X − → ◦ Y may b e rewr itten as ˜ H ϕ ( A, B ) = _ { v ∈ V | ∀ x ∈ A ∃ y ∈ B : v ≤ ϕ ( x, y ) } In this for m the lax functor ˜ H was first considered in [3]. In the prese nce of the Axiom of Choice, so that V is completely distributive in the ordinar y (non-constructive) sense , one ca n then Skolemize the last formula to b eco me ˜ H ϕ ( A, B ) = _ f : A → B ^ x ∈ A ϕ ( x, f ( x )); here the supremum ranges ov er arbitr a ry set mappings f : A → B . Hence, the V W -formula of Lemma 2 has b een tra nscrib ed rather compactly in W V -form. F or the sake of completeness we determine the Eilenberg-Mo ore algebra s o f ˜ H , i.e., those V -catego ries X = ( X, a ) which co me equippe d with a V - mo dule α : H X − → ◦ X s atisfying α · δ X = 1 ∗ X (= a ) ( † ) α · ν X = α · ˜ H α ( ‡ ) The left-hand sides of thos e equa tions are easily computed as ( α · δ X )( x, y ) = _ B ∈ H X δ X ( x, B ) ⊗ α ( B , y ) = _ B ∈ H X h X ( { x } , B ) ⊗ α ( B , y ) = α ( { x } , y ) , ( α · ν X )( A , y ) = _ B ∈ H X ν X ( A , B ) ⊗ α ( B , y ) = _ B ∈ H X h X ( [ A , B ) ⊗ α ( B , y ) = α ( [ A , y ) , ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 17 for all x, y ∈ X , A ∈ H H X . F ur thermore, if k ≤ α ( { x } , x ), for all x ∈ X , then α ( [ A , y ) ≤ ^ A ∈A α ( A, y ) = ˜ H α ( A , { y } ) ⊗ k ≤ ˜ H α ( A , { y } ) ⊗ α ( { y } , y ) ≤ _ B ∈ H X ˜ H α ( A , B ) ⊗ α ( B , y ) = α · ˜ H α ( A , y ) . Consequently , ( † ) and ( ‡ ) imply α ( { x } , y ) = a ( x, y ) and then α ( A, y ) = α  [  { x } | x ∈ A  , y  = ^ x ∈ A α ( { x } , y ) = h X ( A, { y } ) = {− } ∗ ( A, y ) for all A ∈ H X , y ∈ X . Hence, necessarily α = {− } ∗ ; conv ersely , this choice for α satisfies ( † ) and ( ‡ ). Corollary 2. The c ate gory of s t rict ˜ H -algebr as and lax homomorphi sms is the c ate gory V - Mo d itself. Pr o of. A lax homo morphism is, by definition, a V -mo dule ϕ : X − → ◦ Y with ϕ · α ≤ β · ˜ H ϕ (where α, β deno te the uniq ue ly de ter mined structures of X , Y , resp ectively). A straig ht forward calculation shows that e very V -mo dule s atisfies this ine q uality .  6. The Gromo v str ucture for V -ca tegories With ˜ H as in Section 5, one defines GH ( X , Y ) := _ ϕ : X − → ◦ Y ˜ H ϕ ( X , Y ) for all V -c a tegories X and Y . Since for V -functors f : X ′ → X and g : Y ′ → Y o ne has ( g ∗ · ϕ · f ∗ )( x ′ , y ′ ) = ϕ  f ( x ′ ) , g ( y ′ )  for all x ′ ∈ X, y ′ ∈ Y , with Lemma 2 one obta ins immediately GH ( X , Y ) = GH ( X ′ , Y ′ ) whenever f , g are isomor phis ms . Prop ositio n 2. GH is a V - c ate gory structu r e for isomorphism classes of V -c ate gories. Pr o of. Cle arly k ≤ 1 ∗ H X ( X, X ) ≤ ˜ H 1 ∗ X ( X, X ) ≤ GH ( X , X ) , 18 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN and GH ( X , Y ) ⊗ GH ( Y , Z ) = _ ϕ : X − → ◦ Y ,ψ : Y − → ◦ Z ˜ H ϕ ( X , Y ) ⊗ ˜ H ψ ( Y , Z ) ≤ _ ϕ,ψ _ B ∈ H Y ˜ H ϕ ( X , B ) ⊗ ˜ H ψ ( B , Z ) ≤ _ ϕ,ψ ( ˜ H ψ · ˜ H ϕ )( X , Z ) ≤ _ ϕ,ψ ˜ H ( ψ · ϕ )( X, Z ) ≤ _ χ : X − → ◦ Z ˜ H χ ( X , Z ) = GH ( X , Z ) .  W e obser ve that the pr o of relie s on the lax functoriality of ˜ H , but not on the actual definition of ˜ H or H . Hence, instea d of H we may consider any sublifting K : V - Cat → V - Cat of the p owe rset functor , b y whic h we mean an endofunctor K with X ∈ K X ⊆ H X such that the inclusion functions ι X : K X → H X form a la x na tural transforma tion, e.g., they ar e V -functors such tha t f ( A ) = ( H f ) ( A ) ≤ ( K f )( A ) in H Y , for all V - functors f : X → Y and A ∈ K X . (W e have encountered a n example of this situation in Remark s 2(3), with K = H ⇓ .) In this situation w e may r eplace H by K in the pro of o f Prop osition 2 except that for the in v aria nce under isomorphism we inv oked in Lemma 2. But this reference may b e av oided: one easily shows that the diagr ams X ′ y X ′ / / f   c X ′ c f ∗   c X ′ Y ′ y g ∗ · ϕ · f ∗ o o g   X y X / / ˆ X ˆ X c f ∗ O O Y y ϕ o o commute, s o tha t ˜ K ( g ∗ · ϕ · f ∗ ) = ( K y g ∗ · ϕ · f ∗ ) ∗ · ( K y X ′ ) ∗ = ( K g ) ∗ · ( K y ϕ ) ∗ · ( K b f ∗ ) ∗ · ( K y X ′ ) ∗ , while ( K g ) ∗ · ˜ K ϕ · ( K f ) ∗ = ( K g ) ∗ · ( K y ϕ ) ∗ · ( K y X ) ∗ · ( K f ) ∗ = ( K g ) ∗ · ( K y ϕ ) ∗ · ( K c f ∗ ) ∗ · ( K y X ′ ) ∗ . When f is a n isomorphism, one has f − 1 ∗ = f ∗ . Co nsequently , in this case ( K b f ∗ ) ∗ = ( K c f ∗ ) ∗ , and then ˜ K ( g ∗ · ϕ · f ∗ ) = ( K g ) ∗ · ˜ K ϕ · ( K f ) ∗ . ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 19 Hence, when for any sublifting K o f P w e put GK ( X , Y ) := _ ϕ : X − → ◦ Y ˜ K ϕ ( X , Y ) , we may fo r mulate P rop osition 2 more genera lly a s: Theorem 4. GK makes G := ob( V - Cat ) / ∼ = a (lar ge) V -c ate gory, for every su b- lifting K : V - Cat → V - Cat of the p owerset funct or. The resulting V -categor y G K := ( G , GK ) may , with slightly stronger ass umptions on K , b e characterized a s a colimit. F or that purp os e we first prov e: Lemma 3. If K : V - Cat → V - Cat is a 2-functor, then ˜ K ( g ∗ · ϕ · f ∗ ) = ( K g ) ∗ · ˜ K ϕ · ( K f ) ∗ for al l f , g , ϕ as ab ove. Pr o of. It suffices to prov e ( K b f ∗ ) ∗ = ( K c f ∗ ) ∗ for a ll V -functors f : X ′ → X . But since both K a nd the (contrav ar iant) ˆ ( − ) preserve the order of hom-sets, from f ∗ ⊣ f ∗ in V - Mo d we obtain K c f ∗ ⊣ K b f ∗ in V - Cat . Now, since for any pair of V -functor s one has h ⊣ g ⇐ ⇒ g ∗ ⊣ h ∗ ⇐ ⇒ g ∗ = h ∗ , the desired identit y follows with h = K c f ∗ and g = K b f ∗ .  Prop ositio n 3. F or any sublifting K of the p owerset functor pr eserving the or der of hom-sets and ful l fidelity of V -functors one has GK ( X , Y ) = _ X ֒ → Z ← ֓ Y 1 ∗ K Z ( X, Y ) = _ X ֒ → ( X ⊔ Y ,c ) ← ֓Y ˜ K c ( X, Y ) for al l V -c ate gories X and Y . Here the first join ranges ov er all V -categor ies Z into which X a nd Y ma y b e fully embedded, a nd the second o ne ranges over a ll V -c a tegory struc tur es c on the disjoint unio n X ⊔ Y such that X and Y b ecome full V -sub categor ie s. Pr o of. Deno ting the t w o joins b y v , w , r esp ectively , w e trivially hav e w ≤ v , so that v ≤ GK ( X , Y ) ≤ w r emains to b e s hown. Consider ing a ny full e mbedding s X   j X / / Z Y ? _ j Y o o and putting ϕ := j ∗ Y · ( j X ) ∗ = j ∗ Y · 1 ∗ Z · ( j X ) ∗ , beca use of K ’s 2-functor iality a nd preserv ation of full fidelit y we obtain from Lemma 3 a nd Theor em 1 ˜ K ϕ = ( K j Y ) ∗ · ˜ K 1 ∗ Z · ( K j X ) ∗ = j ∗ K Y · 1 ∗ K Z · ( j K X ) ∗ and ther efore 1 ∗ K Z ( X, Y ) = ˜ K ϕ ( X , Y ) ≤ GK ( X, Y ) . Considering any ϕ : X − → ◦ Y , one may define a V -c ategory structure c on X ⊔ Y by c ( z , w ) :=        1 ∗ X ( z , w ) if z , w ∈ X ; ϕ ( z , w ) if z ∈ X , w ∈ Y ; ⊥ if z ∈ Y , w ∈ X ; 1 ∗ Y ( z , w ) if z , w ∈ Y . 20 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN Then, with Z := ( X ⊔ Y , c ), we a gain have ϕ = j ∗ Y · ( j X ) ∗ and o btain ˜ K ϕ ( X, Y ) = ˜ K c ( X, Y ) ≤ w.  Theorem 5. F or K as in Pr op osition 3, G K is a c olimit of the diagr am V - Cat emb / / V - Cat K / / V - Cat   / / V - CA T . Here V - Cat emb is the category of small V -categ ories with full embeddings as morphisms, and V - CA T is the ca tegory of (p ossibly larg e) V -catego ries. Pr o of. The colimit injection κ X : K X → G K sends A ⊆ X to (the isomorphism class of ) A , considered a s a V -categ ory in its own rig ht . Since for A, B ∈ K X one has full embeddings A ֒ → X , B ֒ → X , trivia lly 1 ∗ K X ≤ GK ( A, B ) . Hence κ X is a V -functor, and κ = ( κ X ) X forms a co cone. An y co cone given by V -functor s α X : K X → ( J , J ) allows us to define a V -functor F : G K → J by F X = α X ( X ). Indeed, given V -catego ries X , Y we may consider any Z into which X , Y may b e fully embedded (for e x ample, their copro duct in V - Cat ) and o btain 1 ∗ K Z ( X, Y ) ≤ J  α Z ( X ) , α Z ( Y )  ≤ J  α X ( X ) , α Y ( Y )  = J ( F X , F Y ) . Hence, F is indeed a V -functor with F κ X = α X for all X , and it is ob viously the only such V -functor.  F or the s ake of c o mpleteness we r emark that the as signment K 7→ G K is mo no tone (= functorial): if we or der subliftings o f the p ow erset functor by K ≤ L ⇐ ⇒ th ere is a nat. tr . α : K → L given by inclusio n functions , while V -ca teg ory structure s on G = ob( V - Cat ) / ∼ = carry the p oint wise order (as V -mo dules), then G : Sub H → V - CA T ( G ) bec omes monotone. Indeed, for every V -mo dule ϕ : Z − → ◦ Y , naturality of α gives α ∗ Y · ˜ Lϕ · ( α X ) ∗ = α ∗ Y · ( L y ϕ ) ∗ · ( L y X ) ∗ · ( α X ) ∗ = ( K y ϕ ) ∗ · α ∗ ˆ X · ( α ˆ X ) ∗ · ( K y X ) ∗ ≥ ( K y ϕ ) ∗ · ( K y X ) ∗ = ˜ K ϕ. Consequently , ˜ K ϕ ( X , Y ) ≤ ( α ∗ Y · ˜ Lϕ · ( α X ) ∗ )( X, Y ) = ˜ Lϕ ( α X ( X ) , α Y ( Y )) = ˜ Lϕ ( X , Y ) , which g ives GK ( X , Y ) ≤ GL ( X, Y ), for a ll V -catego ries X , Y . ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 21 7. Opera tions on the Gromo v-Ha usdorff V -ca tegor y Prop ositio n 4. With the binary op er ation ( X , Y ) 7→ X ⊗ Y the V -c ate gory G H b e c omes a monoid in the monoid al c ate gory V - CA T . Pr o of. All w e need to show is that ⊗ : G H ⊗ G H → G H is a V -functor. But for any V -mo dules ϕ : X − → ◦ X ′ , ψ : Y − → ◦ Y ′ and all x ∈ X , y ∈ Y one triv ia lly ha s ˜ H ϕ ( X , X ′ ) ⊗ ˜ H ψ ( Y , Y ′ ) ≤ _ x ′ ∈ X ′ ,y ′ ∈ Y ′ ϕ ( x, x ′ ) ⊗ ψ ( y , y ′ ) , hence ˜ H ϕ ( X , X ′ ) ⊗ ˜ H ψ ( Y , Y ′ ) ≤ ˜ H ( ϕ ⊗ ψ )( X ⊗ Y , X ′ ⊗ Y ′ ) , with the V - mo dule ϕ ⊗ ψ : X ⊗ Y − → ◦ X ′ ⊗ Y ′ given by ( ϕ ⊗ ψ )(( x, y ) , ( x ′ , y ′ )) = ϕ ( x, x ′ ) ⊗ ψ ( y, y ′ ) . Consequently , GH ⊗ GH (( X, Y ) , ( X ′ , Y ′ )) = GH ( X , X ′ ) ⊗ GH ( Y , Y ′ ) = _ ϕ,ψ ˜ H ϕ ( X , X ′ ) ⊗ ˜ H ψ ( Y , Y ′ ) ≤ _ χ : X ⊗ Y − → ◦ X ′ ⊗ Y ′ ˜ H χ ( X ⊗ Y , X ′ ⊗ Y ′ ) = GH ( X ⊗ Y , X ′ ⊗ Y ′ ) .  W e note that when the ⊗ -neutral element k of V is its top elemen t ⊤ , then v ⊗ w ≤ v ∧ w for all v , w ∈ V (since v ⊗ w ≤ v ⊗ k = v ); conv ersely , this ineq ua lit y implies k = ⊤ (since ⊤ = ⊤ ⊗ k ≤ ⊤ ∧ k = k ). Prop ositio n 5. If k = ⊤ in V , then G H b e c omes a monoid in the monoidal c ate gory V - CA T with the binary op er ation give n either by pr o duct or by c opr o duct. Pr o of. W e need to show that × : G H ⊗ G H → G H and + : G H ⊗ G H → G H are b oth V -functor s. Simila r ly to the pro o f of Prop ositio n 4, for the V -functor iality of × it s uffices to show ( § ) ˜ H ϕ ( X , X ′ ) ⊗ ˜ H ψ ( Y , Y ′ ) ≤ ˜ H ( ϕ × ψ )( X × Y , X ′ × Y ′ ) for all V -mo dules ϕ : X − → ◦ X ′ , ψ : Y − → ◦ Y ′ , where ϕ × ψ : X × Y → X ′ × Y ′ is defined by ( ϕ × ψ )(( x, y ) , ( x ′ , y ′ )) = ϕ ( x, x ′ ) ∧ ψ ( y , y ′ ) . (Note that, in this notation, 1 ∗ X × 1 ∗ Y is the V - c ategory structure of the pr o duct X × Y in V - Cat . The verification that ϕ × ψ is indeed a V -mo dule is easy .) But ( § ) follows just like in Prop ositio n 4 since k = ⊤ . F or the V -functoria lit y of + it suffices to establish the inequa lit y ( ¶ ) ˜ H ϕ ( X , X ′ ) ⊗ ˜ H ψ ( Y , Y ′ ) ≤ ˜ H ( ϕ + ψ )( X + Y , X ′ + Y ′ ) , 22 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN with ϕ + ψ : X + Y − → ◦ X ′ + Y ′ defined by ( ϕ + ψ )( z , z ′ ) =    ϕ ( z , z ′ ) if z ∈ X , z ′ ∈ X ′ , ψ ( z , z ′ ) if z ∈ Y , z ′ ∈ Y ′ , ⊥ else. (Again, 1 ∗ X + 1 ∗ Y is precisely the V -categ ory str ucture of the c o pro duct X + Y in V - Cat , a nd the verification of the V -mo dule prop er t y of ϕ + ψ is eas y .) T o v erify ( ¶ ) we consider z ∈ X + Y ; then, for z ∈ X , say , we hav e ˜ H ϕ ( X , X ′ ) ⊗ ˜ H ψ ( Y , Y ′ ) ≤ ˜ H ϕ ( X , X ′ ) ∧ ˜ H ψ ( Y , Y ′ ) ≤ ˜ H ϕ ( X , X ′ ) ≤ _ x ′ ∈ X ′ ϕ ( z , x ′ ) ≤ _ z ′ ∈ X ′ + Y ′ ( ϕ + ψ )( z , z ′ ) , and ( ¶ ) follo ws.  The previous pr o of shows that, without the a ssumption k = ⊤ , one has that + : G H × G H → G H is a V -functor, e.g. that ( G H, +) is a mono id in the Car tesian category V - CA T , but here we will contin ue to consider the monoidal structure of V - CA T . Theorem 6. If k = ⊤ in V , then the Hausdorff fu n ctor H : V - Cat → V - Cat induc es a homomorphism H : ( G H, +) → ( G H, × ) of monoids in the m onoidal c ate gory V - CA T . Pr o of. L e t us first show that the ob ject-part of the functor H : V - Cat → V - Cat defines indeed a V -functor H : G H → G H , so that GH ( X , Y ) ≤ GH ( H X , H Y ) for all V -categ ories X , Y . But for every V -mo dule ϕ : X − → ◦ Y and all A ⊆ X one has ˜ H ϕ ( X, Y ) ≤ ˜ H ϕ ( A, Y ) ≤ _ B ⊆ Y ˜ H ϕ ( A, B ) , which implies ˜ H ϕ ( X , Y ) ≤ ˜ H ( ˜ H ϕ )( H X , H Y ) and then the desir e d inequa lity . In order to identify H a s a ho momorphism, we first note that, as an empty meet, h ∅ ( ∅ , ∅ ) is the top element in V , so that H ∅ ∼ = 1 is terminal in V - Cat , e.g. neutral in ( G H, × ). The bijective map + : H X × H Y → H ( X + Y ) needs to be identified as an isomorphism in V - Cat , e.g . we must show ( h X × h Y )(( A, B ) , ( A ′ , B ′ )) = h X + Y ( A + B , A ′ + B ′ ) for a ll A, A ′ ⊆ X , B , B ′ ⊆ Y . With a = 1 ∗ X and b = 1 ∗ Y , in the notation of the pro of of P rop osition 5 one has _ z ′ ∈ A ′ + B ′ ( a + b )( x, z ′ ) = _ x ′ ∈ A ′ a ( x, x ′ ) ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 23 for all x ∈ A (sinc e ( a + b )( x, z ′ ) = ⊥ when z ′ ∈ B ′ ). Consequently , h X + Y ( A + B , A ′ + B ′ ) = ( ^ x ∈ A _ z ′ ∈ A ′ + B ′ ( a + b )( x, z ′ )) ∧ ( ^ y ∈ B _ z ′ ∈ A ′ + B ′ ( a + b )( y , z ′ )) = ( ^ x ∈ A _ x ′ ∈ A ′ a ( x, x ′ )) ∧ ( ^ y ∈ B _ y ′ ∈ B ′ b ( y , y ′ )) = h X ( A, A ′ ) ∧ h Y ( B , B ′ ) , as desired.  R emarks 4 . (1) T he ( V - Cat )-isomo r phism H X × H Y ∼ = H ( X + Y ) exhibited in the pro o f of Theo rem 6 e a sily extends to the infinite cas e : Y i H X i ∼ = H ( X i X i ) . (2) Since there is no general concept o f a (cov a riant!) functor tra ns forming copro ducts into pro ducts, a mor e enlig h tening ex pla nation for the for m ula just encountered se e ms to b e in order, as follows. Since V - Cat is an exten- sive c ate gory (see [6]), for ev ery (small) family ( X i ) i ∈ I of V -catego ries the functor Σ : Y i V - Cat /X i → V - Cat / X i X i is an equiv alence of categ o ries. Now, the (isomorphism clas ses of a) comma category V - Cat /X can b e made in to a (la rge) V -categ ory when we define the V -c a tegory structur e c b y c ( f , g ) = ^ x ∈ A _ y ∈ B 1 ∗ X ( f ( x ) , g ( y )) = h X ( f ( A ) , g ( B )) , for all f : A → X , g : B → X in V - Cat . In this w ay the equiv alence Σ has b eco me a n isomo r phism o f V - categor ie s, and since H X is just a V -sub categ ory of V - Cat /X , the ( V - Cat )-is omorphism of (1) is simply a re- striction of the iso mo rphism Σ: Q i V - Cat /X i P / / V - Cat / P i X i Q i H X i ?  O O ∼ / / H ( P i X i ) ?  O O 8. Symmetriza tion A V -catego r y X , or just its structure a = 1 ∗ X , is symm et ric when a = a ◦ . This defines the full subca teg ory V - Cat s of V - Cat which is coreflective: the cor eflector sends an ar bitrary X to X s = ( X , a s ) with a s = a × a ◦ , that is: a s ( x, y ) = a ( x, y ) ∧ a ( y , x ) for a ll x, y ∈ X . By H s X = ( H X ) s = ( P X, h s X ) 24 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN one can define a sublifting H s : V - Cat → V - Cat of the p ow erset functor which (like H ) pres erves full fidelit y , but which (unlike H ) fails to b e a 2- functor. How ev er its restriction H s : V - Cat s → V - Cat s is a 2 -functor. R emarks 5 . (1) H s X must not be co nfused with H ( X s ). F or example, for V = 2 and a set X provided with a separ ated (=antisymmetric) o rder, X s carries the dis crete order . Hence, while in H s X one has ( A ≤ B ⇐ ⇒ A ⊆↓ B and B ⊆↓ A ⇐ ⇒ ↓ A = ↓ B ), in H ( X s ) one has ( A ≤ B ⇐ ⇒ A ⊆ B ). (2) E ven after its res triction to V - Cat s there is no easy w ay of e v aluating f H s ϕ ( A, B ) for a V -mo dule ϕ : X − → ◦ Y a nd A ⊆ X , B ⊆ Y , since the com- putation lea ding to the easy formula of Lemma 2 do es not carry throug h when H is repla c ed by H s . (3) T he following a ddendum to Pr op osition 3 suggests how to overcome the difficult y mentioned in (2) when trying to define a no n-trivial symmetric Gromov structur e : V -ca tegory structures c on the disjoint union X ⊔ Y such that the V -catego ries X , Y b ecome full V - s ubca tegories co rresp ond bijectiv ely to pairs of V -mo dules ϕ : X − → ◦ Y , ϕ ′ : Y − → ◦ X with ϕ ′ · ϕ ≤ 1 ∗ X , ϕ · ϕ ′ ≤ 1 ∗ Y ; we write ϕ : X ◦ / / Y : ϕ ′ ◦ o o for such a pair. Under the hypothes e s o f Pr op osition 3 we can no w write GK ( X , Y ) = _ ϕ : X ◦ / / Y : ϕ ′ ◦ o o ˜ K ϕ ( X , Y ) . Hence, for any sublifting K of P we put G s K ( X , Y ) := _ ϕ : X ◦ / / Y : ϕ ′ ◦ o o ˜ K ϕ ( X , Y ) ∧ ˜ K ϕ ′ ( Y , X ) and o btain ea sily: Corollary 3. F or any sublifting K of the p owerset functor, G s K = ( G , G s K ) is a lar ge symmetric V -c ate gory, and when K is a 2-functor pr eserving ful l fidelity of V -functors, then G s K ( X , Y ) = _ X ֒ → Z ← ֓ Y 1 ∗ K Z ( X, Y ) ∧ 1 ∗ K Z ( Y , X ) = _ X ֒ → ( X ⊔ Y ,c ) ← ֓Y ˜ K c ( X, Y ) ∧ ˜ K c ( Y , X ) for al l V -c ate gories X , Y . Pr o of. Rev isiting the proo f of P rop osition 2, we just no te that ϕ : X ◦ / / Y : ϕ ′ ◦ o o , ψ : Y ◦ / / Z : ψ ′ ◦ o o implies ψ · ϕ : X ◦ / / Z : ϕ ′ · ψ ′ ◦ o o . A slight ada ptio n of the computation giv en in Pro po sition 2 now s hows that G s K is indeed a V -catego ry structur e on G = o b V - Cat / ∼ = . The given formulae follow as in the pr o of o f P rop osition 3.  ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 25 Corollary 4. G s H ( X , Y ) = G ( H s )( X, Y ) , for al l V -c ate gories X , Y . Extending the notion of symmetr y from V -c a tegories to V - mo dules, we call a V -mo dule ϕ : X − → ◦ Y symmetric if X, Y are symmetric with ϕ ◦ · ϕ ≤ 1 ∗ X and ϕ · ϕ ◦ ≤ 1 ∗ Y ; we write ϕ : X ◦ / / Y : ϕ ◦ ◦ o o in this s ituation and define G s K ( X , Y ) := _ ϕ : X ◦ / / Y : ϕ ◦ ◦ o o ˜ K ϕ ( X, Y ) for every sublifting K of P . Since s y mmetric V -mo dules comp ose, simila r ly to Corollar y 3 o ne o btains: Corollary 5. F or any sublifting K of the p owerset functor, G s K := (ob V - Cat s / ∼ = , G s K ) is a lar ge V -c ate gory, and when K is a 2-functor pr eserving ful l fidelity of V - functors, then G s K ( X , Y ) = _ X ֒ → Z ← ֓ Y Z symmetric 1 ∗ K Z ( X, Y ) = _ X ֒ → ( X ⊔ Y ,c ) ← ֓Y c symm etric ˜ K c ( X, Y ) for al l symmetric V -c ate gories X , Y .  R emarks 6 . (1) I t is impo rtant to note tha t G s K is not symmetr ic, even when K = H . F or V = P + , X a singleton and Y 3 eq uidistant p oints, we already saw in the Introduction that G s H ( X , Y ) = 0 while G s H ( Y , X ) = 1 2 . Hence it is natura l to conside r the symmetrization ( G s K ) s of G s K : ( G s K ) s ( X, Y ) = G s K ( X , Y ) ∧ G s K ( X , Y ) . The s a me example spaces of the In tro duction show that, while ( GH ) s ( X, Y ) = max { GH ( X , Y ) , GH ( Y , X ) } = 0 , one has ( G s H ) s ( X, Y ) = max { G s H ( X , Y ) , G s H ( Y , X ) } = 1 2 . (2) When the symmetric V -catego ries X , Y are fully embedded in to s ome V - category Z , they ar e a ls o fully embedded into Z s . This fact gives G s H ( X , Y ) ≤ G s H ( X , Y ) which, by symmetry of G s H , gives G s H ( X , Y ) ≤ ( G s H ) s ( X, Y ) . (3) I ns tead o f the co reflector X 7→ X s one may consider the monoidal sym- metrization X sym = ( X , a sym ) with a sym = a ⊗ a ◦ , that is: a sym ( x, y ) = a ( x, y ) ⊗ a ( y , x ). Hence, replacing ∧ by ⊗ one can define H sym X and G sym K in complete a nalogy to H s X a nd G s X , r e s pec tiv ely . Corolla ry 3 r emains v alid when s is tra ded for sy m and ∧ for ⊗ . 26 ANDREI AKHVLEDIANI, MA RIA MANUE L CLEME NTINO, AND W AL TER THOLEN 9. Sep a ra tion A V -categor y X , o r just its structur e a = 1 ∗ X , is sep ar ate d when k ≤ a ( x, y ) ∧ a ( y , x ) implies x = y for a ll x, y ∈ X . It was shown in [10] (and it is easy to verify) that the s eparated V -categ ories form a n epireflective sub catego ry of V - Cat : the image of X under the Y oneda functor y X : X → ˆ X ser ves as the reflecto r. F ur- thermore, there is a closure opera tor which de s crib es separ ation of X equiv alently by the closedness o f the diagonal in X × X . (This description is not needed in what follows, but it fur ther confirms the naturality of the concept.) In Remark s 2 we a lready presented a sublifting H ⇓ of the p ow erset functor , and it is easy to c heck tha t f H ⇓ ϕ ( A, B ) may b e computed as ˜ H ϕ ( A, B ) in Lemma 2, e.g. the t w o v alues coincide, b ecause of the formula proved in Remarks 2(1). F urthermo re, H ⇓ is like H a 2-functor whic h pr eserves full fidelity o f V -functors. Hence, P r op osition 3 is applica ble to H ⇓ and may in fact b e shar p ened to: Corollary 6. F or al l sep ar ate d V -c ate gories X , Y one has GH ( X , Y ) = GH ⇓ ( X, Y ) = _ X ֒ → Z ← ֓ Y Z sep arate d h Z ( X, Y ) = _ X ֒ → ( X ⊔ Y ,c ) ← ֓Y c sep arate d ˜ H c ( X , Y ) . Pr o of. The structure c constructed from a V -mo dule ϕ as in the pro o f of Prop osition 3 is se parated.  R emarks 7 . (1) F rom Coro llary 3 o ne o bta ins G s H ( X , Y ) = G s H ⇓ ( X, Y ) = _ X ֒ → Z ← ֓ Y h Z ( X, Y ) ∧ h Z ( Y , X ) = _ X ֒ → ( X ⊔ Y ,c ) ← ֓Y ˜ H c ( X , Y ) ∧ ˜ H c ( Y , X ) . How e ver, her e it is not p ossible to restric t the last join to separated str uc- tures c : consider the trivial ca s e when V = 2 and X , Y are single ton sets. (2) V -categ ory structures c on X ⊔ Y that are bo th symmetric and separa ted corres p o nd bijectively to sy mmetric mo dules ϕ : X − → ◦ Y with k 6≤ ϕ ( x, y ) for all x ∈ X , y ∈ Y , provided that X and Y ar e b oth s ymmetric and separated. F or V = 2 , X , Y are necessar ily discrete, and the o nly structure c is discrete a s well. (3) T he structur e GH on G is n ot sepa rated, even if we consider only iso mo r- phism classes o f sepa r ated V -catego r ies: for V = 2 , the or der on G given b y GH is c haotic! Likewise when G is traded for G s . References [1] A. Akhvled iani, Hausdorff and Gr omov distanc es in quantale-enriche d c ate gories , M aster’s Thesis, Y ork Unive rsity , 2008. [2] D. Bur ago, Y. Burago and S. Iv anov, A Course in Metric Ge ometry , Ameri can M ath. Society , Providen ce, R. I., 2001. [3] M.M. Clement ino and D. Hofmann, O n extensions of lax monads, The ory Appl. Cate g. 13 (2004) 41–60. [4] M.M. Cl ementino and D. Hofmann, Lawv ere completeness i n topology , Appl. Cate g. Struc- tur es (to appear). [5] M.M. Clementino, D. Hofmann and W. Tholen, One setting for all : metric, top ology , unifor- mity , approac h structure, Appl. Cate g. Structur es 1 2 (2004) 127–154. ON THE CA TEGORICAL MEANING OF HAUSDORFF AND GR OMOV DIST ANCES, I 27 [6] A. Carb oni, S. Lack and R.F.C. W alters, In troduction to extensiv e and distributive categories, J. Pur e Appl. Algebr a 84 (1993) 145–158. [7] M. Gromov, Groups of p olynomial gro wth and expanding maps, Public ations math ´ ematiques de l’I.H.E.S. , 53 (1981) 53-78. [8] M. Gromov, Metric Structur es for Riemannian and Non-Riema nnian Sp ac es , Bir kh¨ auser, Boston, MA, 2007. [9] F. Hausdorff, Grundz¨ uge der Mengenlehr e , T eubner, Leipzig 1914. [10] D. H of mann and W. T holen, La wv ere completion and separation via closure, Appl. Cate g. Structur e s (to appear). [11] G.M . Kelly , Basic c onc epts of enric he d c ate gory the ory , volume 64 of L ondon Mathematic al So ciet y Le ctur e Note Serie s , Cam bridge Univ ersity Pr ess, Camb ridge, 1982. [12] F. W. Lawv ere, Metric spaces, generalized logic, and closed cate gories, R end. Sem. Mat. Fis. Milano 4 3 (1973) 135–166; R eprints in The ory and Applic ations of Cate gories 1 (2002) 1–37. [13] D. Pompeiu, Sur la c ontinuite des fonctions de variables com plexes , Do ctoral thesis, Paris, 1905. [14] I. Stubbe, “Hausdorff distance” via conical co completion, pr eliminary r ep ort , Decem ber 2008. [15] R. J. W o o d, Or dered s ets via adjunctions, i n: Cate g oric al F oundations , volume 97 of Ency- clop e dia Math. Appl. , pages 5–47. Cambridge Univ. Press, Cambridge, 2004. Oxf ord University Computing Labora tor y, Oxford OX1 3QD, Un ited Kin gdom E-mail addr ess : andrei.a khvledia ni@comlab.ox.ac.uk Dep art ment of Ma thema tics , University of Coimb ra, 3 001-454 Coimbra , Por tugal E-mail addr ess : mmc@mat. uc.pt Dep art ment of Ma thema tics and S t a tistics, York University, Toronto, ON M3J 1P3, Canada E-mail addr ess : tholen@m athstat. yorku.ca