On the categorical meaning of Hausdorff and Gromov distances, I
aa r X i v : . [ m a t h . C T ] J a n ON THE CATEGORICAL MEANING OF HAUSDORFFAND GROMOV DISTANCES, I
ANDREI AKHVLEDIANI, MARIA MANUEL CLEMENTINO, AND WALTER THOLEN
Abstract.
Hausdorff and Gromov distances are introduced and treated inthe context of categories enriched over a commutative unital quantale V . TheHausdorff functor which, for every V -category X , provides the powerset of X with a suitable V -category structure, is part of a monad on V - Cat whoseEilenberg-Moore algebras are order-complete. The Gromov construction maybe pursued for any endofunctor K of V - Cat . In order to define the Gromov“distance” between V -categories X and Y we use V -modules between X and Y ,rather than V -category structures on the disjoint union of X and Y . Hence,we first provide a general extension theorem which, for any K , yields a laxextension ˜ K to the category V - Mod of V -categories, with V -modules as mor-phisms. Introduction
The Hausdorff metric for (closed) subsets of a (compact) metric space has beenrecognized for a long time as an important concept in many branches of math-ematics, and its origins reach back even beyond Hausdorff [9], to Pompeiu [13];for a modern account, see [2]. It has gained renewed interest through Gromov’swork [8]. The Gromov-Hausdorff distance of two (compact) metric spaces is theinfimum of their Hausdorff distances after having been isometrically embedded intoany common larger space. There is therefore a notion of convergence for (isometryclasses of compact) metric spaces which has not only become an important toolin analysis and geometry, but which has also provided the key instrument for theproof of Gromov’s existence theorem for a nilpotent subgroup of finite index inevery finitely-generated group of polynomial growth [7].By interpreting the (non-negative) distances d ( x, y ) as hom( x, y ) and, hence, byrewriting the conditions0 ≥ 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 ) , Lawvere [12] described metric spaces as categories enriched over the (small and“thin”) symmetric monoidal-closed category P + = (([0 , ∞ ] , ≥ ) , + , Date : November 1, 2018.2000
Mathematics Subject Classification.
Primary 18E40; Secondary 18A99.The first author acknowledges partial financial assistance from NSERC.The second author acknowledges financial support from the Center of Mathematics of theUniversity of Coimbra/FCT.The third author acknowledges partial financial assistance from NSERC. category ( V , ⊗ , k ). In this paper we present notions of Hausdorff and Gromov dis-tance for the case that V is “thin”. Hence, we replace P + by a commutative andunital quantale V , that is: by a complete lattice which is also a commutative monoid( V , ⊗ , k ) such that the binary operation ⊗ preserves suprema in each variable. Putdifferently, we try to give answers to questions of the type: which structure andproperties of the (extended) non-negative real half-line allow for a meaningful treat-ment of Hausdorff and Gromov distances, and which are their appropriate carriersets? We find that the guidance provided by enriched category theory [11] is almostindispensable for finding satisfactory answers, and that so-called ( bi- ) modules (or distributors ) between V -categories provide an elegant tool for the theory which mayeasily be overlooked without the categorical environment. Hence, our primary mo-tivation for this work is the desire for a better understanding of the true essentialsof the classical metric theory and its applications, rather than the desire for givingmerely a more general framework which, however, may prove to be useful as well.Since ( ∗ ) isolates precisely those conditions of a metric which lend themselvesnaturally to the hom interpretation, a discussion of the others seems to be necessaryat this point; these are: − d ( x, y ) = d ( y, x ) ( symmetry ), − x = y whenever d ( x, y ) = 0 = d ( y, x ) ( separation ), − d ( x, y ) < ∞ ( finiteness ).With the distance of a point x to a subset B of the metric space X = ( X, d ) begiven by d ( x, B ) = inf y ∈ B d ( x, y ), the non-symmetric Hausdorff distance from asubset A to B is defined by Hd ( A, B ) = sup x ∈ A d ( x, B ) , from which one obtains the classical Hausdorff distance H s d ( A, B ) = max { Hd ( A, B ) , Hd ( B, A ) } by enforced symmetrization. But not only symmetry, but also separation and finite-ness get lost under the rather natural passage from d to Hd . (If one thinks of d ( x, B )as the travel time from x to B , then Hd ( A, B ) may be thought of as the time neededto evacuate everyone living in the area A to the area B .) In order to save them oneusually restricts the carrier set from the entire powerset P X to the closed subsetsof X (which makes H s d separated), or even to the non-empty compact subsets(which guarantees also finiteness). As in [10] we call a P + -category an L -metricspace , that is a set X equipped with a function d : X × X → [0 , ∞ ] satisfying ( ∗ );a P + -functor f : ( X, d ) → ( X ′ , d ′ ) is a non-expansive map, e.g. a map f : X → X ′ satisfying d ′ ( f ( x ) , f ( y )) ≤ d ( x, y ) for all x, y ∈ X . That the underlying-set functormakes the resulting category Met topological over
Set (see [5]) provides furthe! revidence that properties ( ∗ ) are fundamental and are better considered separatelyfrom the others, even though symmetry (as a coreflective property) would not ob-struct topologicity. But inclusion of (the reflective property of) separation would,and inclusion of (the neither reflective nor coreflective property of) finiteness wouldmake for an even poorer categorical environment.While symmetry seems to be artificially superimposed on the Hausdorff metric,it plays a crucial role for the Gromov distance, which becomes evident alreadywhen we look at the most elementary examples. Initially nothing prevents us from N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 3 considering arbitrary L -metric spaces X, Y and putting GH ( X, Y ) = inf Z Hd Z ( X, Y ) , where Z runs through all L -metric spaces Z into which both X and Y are isomet-rically embedded. But for X = { p } a singleton set and Y = { x, y, z } three equidis-tant points, with all distances 1, say, for every ε > Z = X ⊔ Y a (proper) metric space, with d ( p, x ) = d ( x, p ) = ε and all other non-zero dis-tances 1. Then Hd 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 separated) structures:put d ( x, p ) = d ( y, p ) = d ( z, p ) = ε , but let the reverse distances be 1. Hence, even aposteriori symmetrization leads to a trivial distance between non-isomorphic spaces.However, there are two ways of a priori symmetrization which both yield the in-tuitively desired result for the Gromov distance in this example: One way is by ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................................................................................................................................................................... .......................................................................................................................................................................................................................................................... x yzp
11 1 restricting the range of the infimum in the definition of GH ( X, Y ) to symmetric L -metric spaces, which seems to be natural when X and Y are symmetric. (Indeed,if for our example spaces one assumes Hd Z ( Y, X ) < with d Z symmetric, thenthe triangle inequality would be violated: 1 ≤ d ( x, p ) + d ( p, y ) < + .) The otherway “works” also for non-symmetric 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 Hd Z ( Y, X ) ≤ , then12 = 1 − ≤ min { d ( p, x ) , d ( p, y ) , d ( p, z ) } = Hd Z ( X, Y ) ≤ H s d Z ( X, Y ) , and when Hd Z ( Y, X ) ≥ , then trivially ≤ H s d Z ( X, Y ).)Having recognized H (and H s ) as endofunctors of Met , these considerationssuggest that G is an “operator” on such endofunctors. But in order to “compute”its values, one needs to “control” the spaces Z in their defining formula, and hereis where modules come in. (A module between L -metric spaces generalizes a non-expansive map just like a relation generalizes a mapping between sets.) A modulefrom X to Y corresponds to an L -metric that one may impose on the disjoint union X ⊔ Y . To take advantage of this viewpoint, it is necessary to extend H from non-expansive maps to modules (leaving its operation on objects unchanged) to becomea lax functor ˜ H . Hence, GH ( X, Y ) may then be more compactly defined using aninfimum that ranges just over the hom-set of modules from X to Y .In Section 2 we give a brief overview of the needed tools from enriched categorytheory, in the highly simplified context of a quantale V . The purpose of Section3 is to establish a general extension theorem for endofunctors of V - Cat , so that
ANDREI AKHVLEDIANI, MARIA MANUEL CLEMENTINO, AND WALTER THOLEN they can act on V -modules rather than just on V -functors. In Sections 4 and 5 weconsider the Hausdorff monad of V - Cat and its lax extension to V -modules, and wedetermine the Eilenberg-Moore category in both cases. The Gromov “distance” isconsidered for a fairly general range of endofunctors of V - Cat in Section 6, and theresulting Gromov “space” of isomorphism classes of V -categories is presented as alarge colimit. For the endofunctor H , in Section 7 this large “space” is shown tocarry internal monoid structures in the monoidal category V - CAT which allow us toconsider H as an internal homomorphism. The effects of symmetrization and thestatus of separation are discussed in Sections 8 and 9. The fundamental questionof transfer of (Cauchy-)completeness from X to HX , as well as the question ofcompleteness of suitable subspaces of the Gromov “space” will be considered in thesecond part of this paper.The reader is reminded that, since P + carries the natural ≥ as its order, in thecontext of a general quantale V the natural infima and suprema of P + appear as joins( W ) and meets ( V ) in V . While this may appear to be irritating initially, it reflects infact the logical viewpoint dictated by the elementary case V = = ( {⊥ < ⊤} , ∧ , ⊤ ),and it translates back well even in the metric case. (For example, if we write thesup-metric d of the real function space C ( X ) as d ( f, g ) = ^ x ∈ X | f ( x ) − g ( x ) | , thenthe statement d ( f, g ) = 0 ⇐⇒ for all x ∈ X : f ( x ) = g ( x )seems to read off the defining formula more directly.) Acknowledgments
While the work presented in this paper first began to take shapewhen, aimed with her knowledge of the treatment of the Hausdorff metric in [3],the second-named author visited the third in the Spring of 2008, which then gaverise to a much more comprehensive study by the first-named author in his Master’sthesis [1] that contains many elements of the current work, precursors of it go infact back to a visit by Richard Wood to the third-named author in 2001. However,the attempt to work immediately with a (non-thin) symmetric monoidal-closedcategory proved to be too difficult at the time. The second- and third-namedauthors also acknowledge encouragement and fertile pointers given by Bill Lawvereover the years, especially after a talk of the third-named author at the Royal FlemishAcademy of Sciences in October 2008. This talk also led to a most interestingexchange with Isar Stubbe who meanwhile has carried the theme of this paper intothe more general context whereby the quantale V is traded for a quantaloid Q (see[14]), a clear indication that the categorical study of the Hausdorff and Gromovmetric may still be in its infancy.2. Quantale-enriched categories
Throughout this paper, V is a commutative, unital quantale. Hence, V is acomplete lattice with a commutative, associative binary operation ⊗ and a ⊗ -neutral element k , such that ⊗ preserves arbitrary suprema in each variable. Ourparadigmatic examples = (cid:0) {⊥ < ⊤} , ∧ , ⊤ (cid:1) and P + = (cid:0) ([0 , ∞ ] , ≥ ) , + , (cid:1) were already considered by Lawvere [12]; they serve to provide both an order-theoretic and a metric intuition for the theory. N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 5 A V -relation r from a set X to a set Y , written as r : X −→7 Y , is simply afunction r : X × Y → V . Its composition with s : Y −→7 Z is given by (cid:0) s · r (cid:1) ( x, z ) = _ y ∈ Y r ( x, y ) ⊗ s ( y, z ) . This defines a category V - Rel , and there is an obvious functor
Set → V - Rel whichassigns to a mapping f : X → Y its V -graph f ◦ : X −→7 Y with f ◦ ( x, y ) = k if f ( x ) = y , and f ◦ ( x, y ) = ⊥ otherwise. This functor is faithful only if k > ⊥ ,which we will assume henceforth, writing just f for f ◦ . There is an involution ( ) ◦ : V - Rel op → V - Rel which sends r : X −→7 Y to r ◦ : Y −→7 X with r ◦ ( y, x ) = r ( x, y ).With the pointwise order of its hom-sets, V - Rel becomes order-enriched, e.g. a2-category, and mappings f : X → Y become maps in the 2-categorical sense:1 X ≤ f ◦ · f, f · f ◦ ≤ Y . A V -category X = ( X, a ) is a set X with a V -relation a : X −→7 X satisfying1 X ≤ a and a · a ≤ a ; elementwise this means 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 equivalently a ( x, y ) ≤ b (cid:0) f ( x ) , f ( y ) (cid:1) for all x, y ∈ X . The resulting category V - Cat yields the category
Ord of (pre)orderedsets and monotone maps for V = and the category Met of L -metric spaces for V = P + . V - Cat has a symmetric monoidal-closed structure, given by(
X, a ) ⊗ ( Y, b ) = ( X × Y, a ⊗ b ) , X – ◦ Y = (cid:0) V - Cat ( X, Y ) , c (cid:1) with a ⊗ b (cid:0) ( x, y ) , ( x ′ , y ′ ) (cid:1) = a ( x, x ′ ) ⊗ b ( y, y ′ ) ,c ( f, g ) = ^ x ∈ X b (cid:0) f ( x ) , g ( x ) (cid:1) . Note that X ⊗ Y must be distinguished from the Cartesian product X × Y whosestructure is a × b with a × b (cid:0) ( x, y ) , ( x ′ , y ′ ) (cid:1) = a ( x, x ′ ) ∧ b ( y, y ′ ) . V itself is a V -category with its “internal hom” – ◦ , given by z ≤ x – ◦ y ⇐⇒ z ⊗ x ≤ y for all x, y, z ∈ V . The morphism → V of quantales has a right adjoint V → that sends v ∈ V to ⊤ precisely when v ≥ k . Hence, there is an induced functor V - Cat → Ord which provides a V -category with the order x ≤ y ⇐⇒ k ≤ a ( x, y ) . Since V - Rel ( X, Y ) = V X × Y = ( X × Y )– ◦V ANDREI AKHVLEDIANI, MARIA MANUEL CLEMENTINO, AND WALTER THOLEN is a V -category (as a product of ( X × Y )-many copies of V , or as a “function space”with X , Y discrete), it is easy to show that V - Rel is (cid:0) V - Cat (cid:1) -enriched, e.g. E → V - Rel ( X, X ) , V - Rel ( X, Y ) ⊗ V - Rel ( Y, Z ) → V - Rel ( X, Z ) ∗ 7→ X , ( r, s ) s · r are V -functors (where E = ( {∗} , k ) is the ⊗ -unit in V - Cat ).3.
Modules, Extension Theorem
For V -categories X = ( X, a ) , Y = ( Y, b ) a V -(bi)module (also: V - distributor , V - profunctor ) ϕ from X to Y , written as ϕ : X −→◦ Y , is a V -relation ϕ : X −→7 Y with ϕ · a ≤ ϕ and 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 all x, x ′ ∈ X, y, y ′ ∈ Y . For ϕ : X −→◦ Y one actually has ϕ · a = ϕ = b · ϕ ,so that 1 ∗ X := a plays the role of the identity morphism in the category V - Mod of V -categories (as objects) and V -modules (as morphisms). It is easy to show that a V -relation ϕ : X −→◦ Y is a V -module 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 - Mod ( X, Y ) = (cid:0) X op ⊗ Y (cid:1) – ◦V . In particular V - Mod is (like V - Rel ) V - Cat - enriched . Also, V - Mod inherits the 2-categorical structure from V - Rel , just via pointwise order.Every V -functor f : X → Y induces adjoint V -modules f ∗ ⊣ f ∗ : Y −→◦ X with f ∗ := b · f , f ∗ := f ◦ · b (in V - Rel ). Hence, there are functors( − ) ∗ : V - Cat → V - Mod , ( − ) ∗ : V - Cat op → V - Mod which map objects identically. V - Cat becomes order-enriched (a 2-category) via f ≤ g ⇐⇒ f ∗ ≤ g ∗ ⇐⇒ ∀ x : f ( x ) ≤ g ( x ) . The V -functor f : X → Y is fully faithful if f ∗ · f ∗ = 1 ∗ X ; equivalently, if a ( x, x ′ ) = b (cid:0) f ( x ) , f ( x ′ ) (cid:1) for all x, x ′ ∈ X .Via ϕ : X −→◦ YX op ⊗ Y → V y ϕ : Y → ( X op – ◦V ) =: ˆ X, every V -module ϕ corresponds to its Yoneda mate y ϕ in V - Cat . In particular, a = 1 ∗ X corresponds to the Yoneda functor y X = y ∗ X : X → ˆ X. For every V -functor f : X op → V and x ∈ X one has1 ∗ ˆ X ( y X ( x ) , f ) = f ( x ) (Yoneda Lemma) . In particular, 1 ∗ ˆ X (cid:0) y X ( x ) , y X ( x ′ ) (cid:1) = a ( x, x ′ ), i.e. y X is fully faithful.The correspondence between ϕ and y ϕ gives: N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 7
Proposition 1. ( − ) ∗ : (cid:0) V - Cat (cid:1) op → V - Mod has a left adjoint ˆ( − ) , given by ˆ ϕ ( s )( x ) = _ y ∈ Y ϕ ( x, y ) ⊗ s ( y ) for all ϕ : X −→◦ Y , s ∈ ˆ Y , x ∈ X .Proof. Under the correspondence 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 adjunction,with η X ( x, t ) = t ( x )for all x ∈ X , t ∈ ˆ X . Note that one has η X = ( y X ) ∗ , by the Yoneda Lemma. Wemust confirm 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 ) = (cid:0) Φ ∗ · ( y X ) ∗ (cid:1) ( x, y )= _ t ∈ ˆ X ∗ ˆ X (cid:0) y X ( x ) , t ) ⊗ ∗ ˆ X (cid:0) t, Φ( y ) (cid:1) ≤ _ t ∈ ˆ X ∗ ˆ X (cid:0) y X ( x ) , Φ( y ) (cid:1) ≤ Φ( y )( x ) ≤ ∗ ˆ X (cid:0) y X ( x ) , y X ( x ) (cid:1) ⊗ ∗ ˆ X (cid:0) y X ( x ) , Φ( y ) (cid:1) ≤ (cid:0) Φ ∗ · ( y X ) ∗ (cid:1) ( x, y )= ϕ ( x, y )for all x ∈ X, y ∈ Y . Hence, necessarily Φ = y ϕ , and the same calculation shows ϕ = ( y ϕ ) ∗ · η X . Now, ˆ ϕ : ˆ Y → ˆ X is the V -functor corresponding to η Y · ϕ , henceˆ ϕ ( s )( x ) = ( η Y · ϕ )( x, s ) = _ y ∈ Y ϕ ( x, y ) ⊗ s ( y ) , for all s ∈ ˆ Y , x ∈ X . (cid:3) Remarks . (1) For ϕ : X −→◦ Y , the V -functor ˆ ϕ may also be described as theleft Kan extension of y ϕ : Y → ˆ X along y Y : Y → ˆ Y .(2) The adjunction of Proposition 1 is in fact 2-categorical. It therefore inducesa 2-monad 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 (cid:0) y, f ( x ) (cid:1) ⊗ t ( x )for t ∈ ˆ X , y ∈ Y . This monad is of Kock-Z¨oberlein type, i.e. one has c y ∗ X ≤ y ˆ X : ˆ X → ˆˆ X. ANDREI AKHVLEDIANI, MARIA MANUEL CLEMENTINO, AND WALTER THOLEN
In fact, for all x, y ∈ X = ( X, a ), and t, s ∈ ˆ X one has a ( x, y ) ≤ s ( x )– ◦ s ( y ),hence t ( y ) ⊗ (cid:0) t ( y )– ◦ a ( x, y ) (cid:1) ⊗ 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 (cid:0) t ( y )– ◦ a ( x, y ) (cid:1) ⊗ s ( x ) ≤ ^ y t ( y )– ◦ s ( y ) = y ˆ X ( s )( t ) . (3) The adjunction of Proposition 1 induces also a monad on V - Mod which wewill not consider further in this paper. But see Section 5 below.(4) Because of (2), the Eilenberg-Moore category( V - Cat ) P V has V -categories X as objects which come equipped with a V -functor α :ˆ X → X with α · y X = 1 X and 1 ˆ X ≤ y X · α , e.g V -categories X for which y X has a left adjoint. These are known to be the V -categories that haveall weighted colimits (see [11]), with α providing a choice of such colimits.Morphisms in ( V - Cat ) P V must preserve the (chosen) weighted colimits.(5) In case V = , P V X can be identified with the set P ↓ X of down-closedsubsets of the (pre)ordered set X , and the Yoneda functor X → P ↓ X sends x to its down-closure ↓ x . Note that P ↓ X is the ordinary power set P X of X when X is discrete. Ord P ↓ has complete ordered sets as objects, andits morphisms must preserve suprema. Hence, this is the category Sup ofso-called sup-lattices (with no anti-symmetry condition).Next we prove a general extension theorem for endofunctors of V - Cat . Whilemaintaining its effect on objects, we wish extend any functor K defined for V -functors to V -modules. To this end we observe that for a V -module ϕ : X −→◦ Y ,the left triangle of ˆ Y ˆ ϕ (cid:31) (cid:31) >>>>>>> ˆ Y ˆ ϕ (cid:31) (cid:31) >>>>>>> Y y Y @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) y ϕ / / ˆ X Z y ψ @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) y ψ · ϕ / / ˆ X commutes, since y Y is the counit of the adjunction of Proposition 1. More generally,the right triangle commutes for every ψ : Y −→◦ Z . Theorem 1 (Extension Theorem) . For every functor K : V - Cat → V - Cat , ˜ Kϕ := (cid:0) KX ◦ ( K y X ) ∗ / / K ˆ X ◦ ( K y ϕ ) ∗ / / KY (cid:1) N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 9 defines a lax functor ˜ K : V - Mod → V - Mod which coincides with K on objects.Moreover, if K preserves full fidelity of V -functors, the diagram V - Mod ˜ K / / V - Mod ( V - Cat ) op( − ) ∗ O O K op / / ( V - Cat ) op( − ) ∗ O O commutes.Proof. Lax functoriality of ˜ K follows from˜ K (1 ∗ X ) = ( K y X ) ∗ · ( K y X ) ∗ ≥ ∗ KX , ˜ 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 ) ∗ = ˜ Kψ · ˜ Kϕ, 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 adjunction.For a V -functor f : X → Y , the triangle Y y Y @@@@@@@ X f ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) y f ∗ / / ˆ Y . commutes, so that˜ K ( f ∗ ) = ( K y f ∗ ) ∗ · ( K y Y ) ∗ = ( Kf ) ∗ ( K y Y ) ∗ ( K y Y ) ∗ ≥ ( Kf ) ∗ , and one even has ˜ K ( f ∗ ) = ( Kf ) ∗ if K preserves the full fidelity of y Y . (cid:3) The Hausdorff Monad on V - Cat
Let X = ( X, a ) be a V -category. Then ˆ X = ( X op – ◦V ) = P V X is closed undersuprema formed in the product V X ; hence, like V it is a sup-lattice. Consequently,the Yoneda 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 ! ! DDDDDDDD
P X Y X (cid:15) (cid:15) B _ (cid:15) (cid:15) P V X a ( − , B )where a ( x, B ) = _ y ∈ B a ( x, y )for all x ∈ X , B ⊆ X . We can provide the set P X with a V -category structure h X which it inherits from P V X (since the forgetful functor V - Cat → Set is a fibration, even a topological functor, see [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 ) . Proof.
From k ≤ a ( x, A ) for all x ∈ A one obtains h X ( A, B ) ≤ ^ x ∈ A a ( x, A )– ◦ a ( x, B ) ≤ ^ x ∈ A k – ◦ a ( x, B ) = ^ x ∈ A a ( x, B ) . Conversely, with v := ^ x ∈ A _ y ∈ B a ( x, y ), we must show v ≤ a ( z, A ) – ◦ a ( z, B ) for all z ∈ X . But since for every x ∈ Aa ( 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 desired. (cid:3) For a V -functor f : X → Y = ( Y, b ) one now concludes easily h X ( A, B ) ≤ ^ x ∈ A _ y ∈ B b (cid:0) f ( x ) , f ( y ) (cid:1) = h Y (cid:0) f ( A ) , f ( B ) (cid:1) for all A, B ⊆ X . Consequently, with HX = ( P X, h X ) , Hf : HX → HY, A f ( A ) , one obtains a (2-)functor H which makes the diagram V - Cat H / / (cid:15) (cid:15) V - Cat (cid:15) (cid:15)
Set P / / Set commute. Actually, one has the following theorem:
Theorem 2.
The powerset monad P = ( P, {−} , S ) can be lifted along the forgetfulfunctor V - Cat → Set to a monad H of V - Cat of Kock-Z¨oberlein type.Proof.
For a V -category X , x
7→ { x } gives a fully faithful V -functor {−} : X → HX . In order to show that [ : HHX → HX,
A 7→ [ A , is a V -functor, it suffices to verify that for all x ∈ A ∈ A ∈ HHX and
B ∈
HHX one has h HX ( 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 HX ( A , B ) ≤ _ B ∈B h X ( A, B ) ≤ a ( x, [ B ) . N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 11
The induced order of HX is given by A ≤ B ⇐⇒ ∀ x ∈ A : k ≤ a ( x, B ) , and that of HHX by A ≤ B ⇐⇒ ∀ A ∈ A : k ≤ _ B ∈B h X ( A, B ) . Hence, from k ≤ a ( x, A ) = h X ( { x } , A ) for all A ∈ HX one obtains (cid:8) { x } | x ∈ A (cid:9) ≤ { A } in HHX , which means H {−} X ≤ {−} HX , i.e., H is Kock-Z¨oberlein. (cid:3) Remarks . (1) By definition, h X ( A, B ) depends only on a ( − , A ), a ( − , B ).Hence, if we put ⇓ X B := (cid:8) x ∈ X | { x } ≤ B (cid:9) = { x ∈ X | ↓ x ≤ B } = { x ∈ X | k ≤ a ( x, B ) } , from B ⊆⇓ X B one trivially has a ( z, B ) ≤ a ( z, ⇓ X B ) for all z ∈ X , butalso 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) must not be confused with the down-closure ↓ X B of B in X w.r.t the induced order of X , e.g. with ↓ X B = { x ∈ X | ∃ y ∈ B x ≤ y } = { x ∈ X | ∃ y ∈ B ( k ≤ a ( x, y )) } . In general, B ⊆↓ X B ⊆⇓ X B . While ↓ X B = ⇓ X B for V = , the two setsare generally distinct even for V = P + .(3) In the induced order of HX one has A ≤ B ⇐⇒ A ⊆⇓ X B. Hence, if we restrict HX to H ⇓ X := { B ⊆ X | B = ⇓ X B } , the induced order of H ⇓ X is simply the inclusion order. H ⇓ becomes afunctor H ⇓ : V - Cat → V - Cat with( H ⇓ f )( A ) = ⇓ Y f ( A )for all A ∈ H ⇓ X , and there is a lax natural transformation ι : H ⇓ → H given by inclusion functions. Like H , also H ⇓ carries a monad structure,given by X → H ⇓ X, x X x = ⇓ X x,H ⇓ H ⇓ X → H ⇓ X, B 7→⇓ X ( [ B ) . In this way ι : H ⇓ → H becomes a lax monad morphism. (4) By definition, y X is fully faithful. Hence, HX carries the largest V -categorystructure making y X : HX → P V X a V -functor. Equivalently, this is thelargest V -category structure making δ X : X −→◦ HX with δ ( x, B ) = a ( x, B ) a V -module.(5) Y X : HX → P V X defines a morphism H → P V of monads. Indeed, the leftdiagram of X {−} } } {{{{{{{{ y X ! ! DDDDDDDD
HHX S (cid:15) (cid:15) H Y X / / HP V X Y ˆ X / / P V P V X m X (cid:15) (cid:15) HX Y X / / P V X HX Y X / / P V X commutes trivially, and for the right one first observes that m X : ˆˆ 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 ∈ ˆˆ X , x ∈ X . Hence, for B ∈
HHX we have:( m X · Y ˆ X · H Y X )( B )( x ) = _ t ∈ ˆ X t ( x ) ⊗ Y ˆ X (cid:0) Y X ( B ) (cid:1) = _ t ∈ ˆ X t ( x ) ⊗ ∗ ˆ X (cid:0) t, Y X ( B ) (cid:1) = _ t ∈ ˆ X _ B ∈B t ( x ) ⊗ (cid:0) ^ x ′ ∈ X t ( x ′ )– ◦ a ( x ′ , B ) (cid:1) ≤ _ B ∈B _ t ∈ ˆ X t ( x ) ⊗ (cid:0) t ( x )– ◦ a ( x, B ) (cid:1) = _ B ∈B a ( x, B )= Y X ( [ B )( x ) ≤ a ( x, x ) ⊗ _ B ∈B ∗ ˆ X (cid:0) y X ( x ) , a ( − , B ) (cid:1) ≤ _ t ∈ ˆ X t ( x ) ⊗ ∗ ˆ X (cid:0) t, Y X ( B ) (cid:1) = ( m X · Y ˆ X · H Y X )( B )( x ) . Consequently, there is an induced algebraic functor( V - Cat ) H → ( V - Cat ) P V of the respective Eilenberg-Moore categories.We briefly describe the Eilenberg-Moore category( V - Cat ) H N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 13 where objects X ∈ V - Cat come equipped with a V -functor α : HX → X satisfying α · {−} = 1 X and 1 HX ≤ {−} · α (since H is Kock-Z¨oberlein). Hence, α ( { x } ) = x for all x ∈ X , and A ≤ { α ( A ) } for A ∈ HX , that is: k ≤ h X ( A, { α ( A ) } ) = ^ x ∈ A a (cid:0) x, α ( A ) (cid:1) . Consequently, α ( A ) is an upper bound of A in the induced order of X , and for anyother upper bound y of A in X = ( X, a ) one has k ≤ ^ x ∈ A a ( x, y ) = h X ( A, { y } ) ≤ a (cid:0) α ( A ) , α ( { y } ) (cid:1) = a (cid:0) α ( A ) , y (cid:1) since α is a V -functor. Hence, α ( A ) gives (a choice of) a supremum of A in X .Moreover, the last computation shows( ∗ ) a ( _ A, y ) = ^ x ∈ A a ( x, y )for all y ∈ X, A ∈ HX (since “ ≤ ” holds trivially). Conversely, any V -category X = ( X, a ) which is complete in its induced order and satisfies ( ∗ ) is easily seen tobe an object of ( V - Cat ) H . Corollary 1.
The Eilenberg-Moore category of H has order-complete V -categories X = ( X, a ) satisfying ( ∗ ) as its objects, and morphisms are V -functors preserving(the chosen) suprema. The lax Hausdorff monad on V - Mod
When applying Theorem 1 to the Hausdorff functor H : V - Cat → V - Cat ofTheorem 2 we obtain a lax functor ˜ H : V - Mod → V - Mod whose value on a V -module ϕ : X −→◦ Y may be easily computed: Lemma 2. ˜ Hϕ ( A, B ) = ^ x ∈ A _ y ∈ B ϕ ( x, y ) for all subsets A ⊆ X, B ⊆ Y . Proof. ˜ Hϕ ( A, B ) = _ D ∈ H ˆ X ( H y X ) ∗ ( A, D ) ⊗ ( H y ϕ ) ∗ ( D, B )= _ D ∈ H ˆ X h ˆ X (cid:0) y X ( A ) , D (cid:1) ⊗ h ˆ X (cid:0) D, y ϕ ( B ) (cid:1) ≤ h ˆ X (cid:0) y X ( A ) , y ϕ ( B ) (cid:1) = ^ x ∈ A _ y ∈ B ∗ ˆ X (cid:0) y X ( x ) , y ϕ ( y ) (cid:1) = ^ x ∈ A _ y ∈ B ϕ ( x, y ) ( Y oneda )= h ˆ X (cid:0) y X ( A ) , y ϕ ( B ) (cid:1) ≤ h ˆ X (cid:0) y X ( A ) , y X ( A ) (cid:1) ⊗ h ˆ X (cid:0) y X ( A ) , y ϕ ( B ) (cid:1) ≤ _ D ∈ H ˆ X h ˆ X (cid:0) y X ( A ) , D (cid:1) ⊗ h ˆ X (cid:0) D, y ϕ ( B ) (cid:1) = ˜ Hϕ ( A, B ) . (cid:3) We now prove that ˜ H carries a lax monad structure. Theorem 3. ˜ H belongs to a lax monad ˜ H = ( ˜ H, δ, ν ) of V - Mod such that H ofTheorem 2 is a lifting of ˜ H along ( − ) ∗ : V - Cat → V - Mod .Proof.
Let us first note that H is a lifting of ˜ H along ( − ) ∗ , in the sense that V - Cat H / / ( − ) ∗ (cid:15) (cid:15) V - Cat ( − ) ∗ (cid:15) (cid:15) V - Mod ˜ H / / V - Mod commutes. Indeed, for f : X → Y = ( Y, b ) in V - Cat and A ∈ HX , B ∈ HY onehas ˜ H ( f ∗ )( A, B ) = ^ x ∈ A _ y ∈ B b (cid:0) f ( x ) , y (cid:1) = h Y (cid:0) f ( A ) , B ) (cid:1) = ( Hf ) ∗ ( A, B ) . The unit of ˜ H , δ : 1 → ˜ H , is defined by δ X = {−} ∗ : X −→◦ HX, δ X ( x, B ) = h X ( { x } , B ) = a ( x, B ) , for X = ( X, a ), x ∈ X , B ∈ HX (see also Remarks 2 (2)), and the multiplication ν : ˜ H ˜ H → ˜ H can be given by ν X = [ ∗ : HHX −→◦
HX, ν X ( A , B ) = h X ( [ A , B ) = ^ A ∈A h X ( A, B ) , for A ∈
HHX , B ∈ HX . The monad conditions hold strictly for ˜ H , because theyhold strictly for H . For example, ν · ˜ Hδ = 1 follows from ν X · ˜ Hδ X = [ ∗ · ˜ H ( {−} ∗ ) = [ ∗ · ( H {−} ) ∗ = ( [ · H {−} ) ∗ = 1 ∗ X . N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 15
Surprisingly though, also the naturality squares for both δ X and ν X commute strictly . Indeed, for ϕ : X −→◦ Y = ( Y, b ), x ∈ X , B ∈ HY and A ∈
HHX one has:( ˜ Hϕ · δ X )( x, B ) = _ A ∈ HX δ X ( x, A ) ⊗ ˜ Hϕ ( A, B )= _ A ∈ HX h X ( { x } , A ) ⊗ ˜ Hϕ ( A, B )= ˜ Hϕ ( { x } , B )= _ y ∈ B ϕ ( x, y )= _ y ∈ B _ z ∈ Y ϕ ( x, z ) ⊗ b ( z, y )= _ z ∈ Y ϕ ( x, z ) ⊗ (cid:0) _ y ∈ B b ( z, y ) (cid:1) = _ z ∈ Y ϕ ( x, z ) ⊗ δ Y ( z, B )= ( δ Y · ϕ )( x, B ) , ( ˜ Hϕ · ν X )( A , B ) = _ A ∈ HX ν X ( A , A ) ⊗ ˜ Hϕ ( A, B )= _ A ∈ HX h X ( [ A , A ) ⊗ ˜ Hϕ ( A, B )= ˜ Hϕ ( [ A , B ) ≤ (cid:0) ^ A ∈A _ B ′ ∈ HB ˜ Hϕ ( A, B ′ ) (cid:1) ⊗ ^ B ′ ∈ HB h Y ( B ′ , B )(since k ≤ h Y ( B ′ , B ) for B ′ ∈ HB ) ≤ _ B∈ HHY (cid:0) ^ A ∈A _ B ′ ∈B ˜ Hϕ ( A, B ′ ) (cid:1) ⊗ (cid:0) ^ B ′ ∈B h Y ( B ′ , B ) (cid:1) = ( ν Y · ˜ H ˜ Hϕ )( A , B )= _ B∈ HHY ˜ H ˜ Hϕ ( A , B ) ⊗ ν Y ( B , B ) ≤ _ B∈ HHY ^ A ∈A ˜ Hϕ ( A, [ B ) ⊗ h Y ( [ B , B ) ≤ ^ A ∈A ˜ Hϕ ( A, B )= ( ˜ Hϕ · ν X )( A , B ) . (cid:3) Remarks . (1) We emphasize that, while ˜ H is only a lax functor, this is infact the only defect that prevents ˜ H from being a monad in the strict sense.(2) In addition to the commutativity of the diagram given in the Proof ofTheorem 3, since H obviously preserves full fidelity of V -functors, from Theorem 1 we obtain also the commutativity of( V - Cat ) op H op / / ( − ) ∗ (cid:15) (cid:15) ( V - Cat ) op( − ) ∗ (cid:15) (cid:15) V - Mod ˜ H / / V - Mod (3) If V is constructively completely distributive (see [15], [3]), then ˜ Hϕ for ϕ : X −→◦ Y may be rewritten as˜ Hϕ ( A, B ) = _ { v ∈ V | ∀ x ∈ A ∃ y ∈ B : v ≤ ϕ ( x, y ) } In this form the lax functor ˜ H was first considered in [3]. In the presenceof the Axiom of Choice, so that V is completely distributive in the ordinary(non-constructive) sense, one can then Skolemize the last formula to become˜ Hϕ ( A, B ) = _ f : A → B ^ x ∈ A ϕ ( x, f ( x ));here the supremum ranges over arbitrary set mappings f : A → B . Hence,the V W -formula of Lemma 2 has been transcribed rather compactly in
W V -form.For the sake of completeness we determine the Eilenberg-Moore algebras of ˜ H ,i.e., those V -categories X = ( X, a ) which come equipped with a V -module α : HX −→◦ X satisfying α · δ X = 1 ∗ X (= a )( † ) α · ν X = α · ˜ Hα ( ‡ )The left-hand sides of those equations are easily computed as( α · δ X )( x, y ) = _ B ∈ HX δ X ( x, B ) ⊗ α ( B, y )= _ B ∈ HX h X ( { x } , B ) ⊗ α ( B, y )= α ( { x } , y ) , ( α · ν X )( A , y ) = _ B ∈ HX ν X ( A , B ) ⊗ α ( B, y )= _ B ∈ HX h X ( [ A , B ) ⊗ α ( B, y )= α ( [ A , y ) , N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 17 for all x, y ∈ X , A ∈
HHX . Furthermore, if k ≤ α ( { x } , x ), for all x ∈ X , then α ( [ A , y ) ≤ ^ A ∈A α ( A, y )= ˜ Hα ( A , { y } ) ⊗ k ≤ ˜ Hα ( A , { y } ) ⊗ α ( { y } , y ) ≤ _ B ∈ HX ˜ Hα ( A , B ) ⊗ α ( B, y )= α · ˜ Hα ( A , y ) . Consequently, ( † ) and ( ‡ ) imply α ( { x } , y ) = a ( x, y ) and then α ( A, y ) = α (cid:16) [ (cid:8) { x } | x ∈ A (cid:9) , y (cid:17) = ^ x ∈ A α ( { x } , y )= h X ( A, { y } ) = {−} ∗ ( A, y )for all A ∈ HX , y ∈ X . Hence, necessarily α = {−} ∗ ; conversely, this choice for α satisfies ( † ) and ( ‡ ). Corollary 2.
The category of strict ˜ H -algebras and lax homomorphisms is thecategory V - Mod itself.Proof.
A lax homomorphism is, by definition, a V -module ϕ : X −→◦ Y with ϕ · α ≤ β · ˜ Hϕ (where α, β denote the uniquely determined structures of X , Y , respectively).A straightforward calculation shows that every V -module satisfies this inequality. (cid:3) The Gromov structure for V -categories With ˜ H as in Section 5, one defines GH ( X, Y ) := _ ϕ : X −→◦ Y ˜ Hϕ ( X, Y )for all V -categories X and Y . Since for V -functors f : X ′ → X and g : Y ′ → Y onehas ( g ∗ · ϕ · f ∗ )( x ′ , y ′ ) = ϕ (cid:0) f ( x ′ ) , g ( y ′ ) (cid:1) for all x ′ ∈ X, y ′ ∈ Y , with Lemma 2 one obtains immediately GH ( X, Y ) = GH ( X ′ , Y ′ )whenever f , g are isomorphisms. Proposition 2. GH is a V -category structure for isomorphism classes of V -categories.Proof. Clearly k ≤ ∗ HX ( X, X ) ≤ ˜ H ∗ X ( X, X ) ≤ GH ( X, X ) , and GH ( X, Y ) ⊗ GH ( Y, Z ) = _ ϕ : X −→◦ Y,ψ : Y −→◦ Z ˜ Hϕ ( X, Y ) ⊗ ˜ Hψ ( Y, Z ) ≤ _ ϕ,ψ _ B ∈ HY ˜ Hϕ ( X, B ) ⊗ ˜ Hψ ( B, Z ) ≤ _ ϕ,ψ ( ˜ Hψ · ˜ Hϕ )( X, Z ) ≤ _ ϕ,ψ ˜ H ( ψ · ϕ )( X, Z ) ≤ _ χ : X −→◦ Z ˜ Hχ ( X, Z )= GH ( X, Z ) . (cid:3) We observe that the proof relies on the lax functoriality of ˜ H , but not on theactual definition of ˜ H or H . Hence, instead of H we may consider any sublifting K : V - Cat → V - Cat of the powerset functor , by which we mean an endofunctor K with X ∈ KX ⊆ HX such that the inclusion functions ι X : KX → HX form a lax natural transformation, e.g., they are V -functors such that f ( A ) = ( Hf )( A ) ≤ ( Kf )( A )in HY , for all V -functors f : X → Y and A ∈ KX . (We have encountered anexample of this situation in Remarks 2(3), with K = H ⇓ .) In this situation wemay replace H by K in the proof of Proposition 2 except that for the invarianceunder isomorphism we invoked in Lemma 2. But this reference may be avoided:one easily shows that the diagrams X ′ y X ′ / / f (cid:15) (cid:15) c X ′ c f ∗ (cid:15) (cid:15) c X ′ Y ′ y g ∗· ϕ · f ∗ o o g (cid:15) (cid:15) X y X / / ˆ X ˆ X c f ∗ O O Y y ϕ o o commute, so that˜ K ( g ∗ · ϕ · f ∗ ) = ( K y g ∗ · ϕ · f ∗ ) ∗ · ( K y X ′ ) ∗ = ( Kg ) ∗ · ( K y ϕ ) ∗ · ( K b f ∗ ) ∗ · ( K y X ′ ) ∗ , while ( Kg ) ∗ · ˜ Kϕ · ( Kf ) ∗ = ( Kg ) ∗ · ( K y ϕ ) ∗ · ( K y X ) ∗ · ( Kf ) ∗ = ( Kg ) ∗ · ( K y ϕ ) ∗ · ( K c f ∗ ) ∗ · ( K y X ′ ) ∗ . When f is an isomorphism, one has f − ∗ = f ∗ . Consequently, in this case ( K b f ∗ ) ∗ =( K c f ∗ ) ∗ , and then ˜ K ( g ∗ · ϕ · f ∗ ) = ( Kg ) ∗ · ˜ Kϕ · ( Kf ) ∗ . N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 19
Hence, when for any sublifting K of P we put GK ( X, Y ) := _ ϕ : X −→◦ Y ˜ Kϕ ( X, Y ) , we may formulate Proposition 2 more generally as: Theorem 4. GK makes G := ob( V - Cat ) / ∼ = a (large) V -category, for every sub-lifting K : V - Cat → V - Cat of the powerset functor.
The resulting V -category G K := ( G , GK )may, with slightly stronger assumptions on K , be characterized as a colimit. Forthat purpose we first prove: Lemma 3. If K : V - Cat → V - Cat is a 2-functor, then ˜ K ( g ∗ · ϕ · f ∗ ) = ( Kg ) ∗ · ˜ Kϕ · ( Kf ) ∗ for all f, g, ϕ as above.Proof. It suffices to prove ( K b f ∗ ) ∗ = ( K c f ∗ ) ∗ for all V -functors f : X ′ → X . Butsince both K and the (contravariant) ˆ( − ) preserve the order of hom-sets, from f ∗ ⊣ f ∗ in V - Mod we obtain K c f ∗ ⊣ K b f ∗ in V - Cat . Now, since for any pair of V -functors one has h ⊣ g ⇐⇒ g ∗ ⊣ h ∗ ⇐⇒ g ∗ = h ∗ , the desired identity follows with h = K c f ∗ and g = K b f ∗ . (cid:3) Proposition 3.
For any sublifting K of the powerset functor preserving the orderof hom-sets and full fidelity of V -functors one has GK ( X, Y ) = _ X֒ → Z ← ֓Y ∗ KZ ( X, Y ) = _ X֒ → ( X ⊔ Y,c ) ← ֓Y ˜ Kc ( X, Y ) for all V -categories X and Y . Here the first join ranges over all V -categories Z into which X and Y may befully embedded, and the second one ranges over all V -category structures c on thedisjoint union X ⊔ Y such that X and Y become full V -subcategories. Proof.
Denoting the two joins by v , w , respectively, we trivially have w ≤ v , sothat v ≤ GK ( X, Y ) ≤ w remains to be shown. Considering any full embeddings X (cid:31) (cid:127) j X / / Z Y ? _ j Y o o and putting ϕ := j ∗ Y · ( j X ) ∗ = j ∗ Y · ∗ Z · ( j X ) ∗ , because of K ’s 2-functoriality andpreservation of full fidelity we obtain from Lemma 3 and Theorem 1˜ Kϕ = ( Kj Y ) ∗ · ˜ K ∗ Z · ( Kj X ) ∗ = j ∗ KY · ∗ KZ · ( j KX ) ∗ and therefore 1 ∗ KZ ( X, Y ) = ˜ Kϕ ( X, Y ) ≤ GK ( X, Y ) . Considering any ϕ : X −→◦ Y , one may define a V -category structure c on X ⊔ Y by c ( z, w ) := ∗ 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 . Then, with Z := ( X ⊔ Y, c ), we again have ϕ = j ∗ Y · ( j X ) ∗ and obtain˜ Kϕ ( X, Y ) = ˜ Kc ( X, Y ) ≤ w. (cid:3) Theorem 5.
For K as in Proposition 3, G K is a colimit of the diagram V - Cat emb / / V - Cat K / / V - Cat (cid:31) (cid:127) / / V - CAT . Here V - Cat emb is the category of small V -categories with full embeddings asmorphisms, and V - CAT is the category of (possibly large) V -categories. Proof.
The colimit injection κ X : KX → G K sends A ⊆ X to (the isomorphismclass of) A , considered as a V -category in its own right. Since for A, B ∈ KX onehas full embeddings A ֒ → X, B ֒ → X , trivially1 ∗ KX ≤ GK ( A, B ) . Hence κ X is a V -functor, and κ = ( κ X ) X forms a cocone. Any cocone given by V -functors α X : KX → ( J , J ) allows us to define a V -functor F : G K → J by F X = α X ( X ). Indeed, given V -categories X, Y we may consider any Z into which X, Y may be fully embedded (for example, their coproduct in V - Cat ) and obtain1 ∗ KZ ( X, Y ) ≤ J (cid:0) α Z ( X ) , α Z ( Y ) (cid:1) ≤ J (cid:0) α X ( X ) , α Y ( Y ) (cid:1) = J ( F X, F Y ) . Hence, F is indeed a V -functor with F κ X = α X for all X , and it is obviously theonly such V -functor. (cid:3) For the sake of completeness we remark that the assignment K
7→ G K is monotone (=functorial): if we order subliftings of the powerset functor by K ≤ L ⇐⇒ there is a nat. tr. α : K → L given by inclusion functions , while V -category structures on G = ob( V - Cat ) / ∼ = carry the pointwise order (as V -modules), then G : Sub H → V - CAT ( G )becomes monotone. Indeed, for every V -module 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 ) ∗ = ˜ Kϕ.
Consequently, ˜ Kϕ ( X, Y ) ≤ ( α ∗ Y · ˜ Lϕ · ( α X ) ∗ )( X, Y )= ˜ Lϕ ( α X ( X ) , α Y ( Y ))= ˜ Lϕ ( X, Y ) , which gives GK ( X, Y ) ≤ GL ( X, Y ), for all V -categories X, Y . N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 21 Operations on the Gromov-Hausdorff V -category Proposition 4.
With the binary operation ( X, Y ) X ⊗ Y the V -category G H becomes a monoid in the monoidal category V - CAT .Proof.
All we need to show is that ⊗ : G H ⊗ G H → G H is a V -functor. But for any V -modules ϕ : X −→◦ X ′ , ψ : Y −→◦ Y ′ and all x ∈ X , y ∈ Y one trivially has˜ Hϕ ( X, X ′ ) ⊗ ˜ Hψ ( Y, Y ′ ) ≤ _ x ′ ∈ X ′ ,y ′ ∈ Y ′ ϕ ( x, x ′ ) ⊗ ψ ( y, y ′ ) , hence ˜ Hϕ ( X, X ′ ) ⊗ ˜ Hψ ( Y, Y ′ ) ≤ ˜ H ( ϕ ⊗ ψ )( X ⊗ Y, X ′ ⊗ Y ′ ) , with the V -module 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 ˜ Hϕ ( X, X ′ ) ⊗ ˜ Hψ ( Y, Y ′ ) ≤ _ χ : X ⊗ Y −→◦ X ′ ⊗ Y ′ ˜ Hχ ( X ⊗ Y, X ′ ⊗ Y ′ )= GH ( X ⊗ Y, X ′ ⊗ Y ′ ) . (cid:3) We note that when the ⊗ -neutral element k of V is its top element ⊤ , then v ⊗ w ≤ v ∧ w for all v, w ∈ V (since v ⊗ w ≤ v ⊗ k = v ); conversely, this inequalityimplies k = ⊤ (since ⊤ = ⊤ ⊗ k ≤ ⊤ ∧ k = k ). Proposition 5. If k = ⊤ in V , then G H becomes a monoid in the monoidalcategory V - CAT with the binary operation given either by product or by coproduct.Proof.
We need to show that × : G H ⊗ G H → G H and + : G H ⊗ G H → G H are both V -functors. Similarly to the proof of Proposition 4, for the V -functorialityof × it suffices to show( § ) ˜ Hϕ ( X, X ′ ) ⊗ ˜ Hψ ( Y, Y ′ ) ≤ ˜ H ( ϕ × ψ )( X × Y, X ′ × Y ′ )for all V -modules ϕ : X −→◦ X ′ , ψ : Y −→◦ Y ′ , where ϕ × ψ : X × Y → X ′ × Y ′ isdefined by ( ϕ × ψ )(( x, y ) , ( x ′ , y ′ )) = ϕ ( x, x ′ ) ∧ ψ ( y, y ′ ) . (Note that, in this notation, 1 ∗ X × ∗ Y is the V -category structure of the product X × Y in V - Cat . The verification that ϕ × ψ is indeed a V -module is easy.) But ( § )follows just like in Proposition 4 since k = ⊤ .For the V -functoriality of + it suffices to establish the inequality( ¶ ) ˜ Hϕ ( X, X ′ ) ⊗ ˜ Hψ ( Y, Y ′ ) ≤ ˜ H ( ϕ + ψ )( X + Y, X ′ + Y ′ ) , 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 -category structure of the coproduct X + Y in V - Cat , and the verification of the V -module property of ϕ + ψ is easy.) To verify( ¶ ) we consider z ∈ X + Y ; then, for z ∈ X , say, we have˜ Hϕ ( X, X ′ ) ⊗ ˜ Hψ ( Y, Y ′ ) ≤ ˜ Hϕ ( X, X ′ ) ∧ ˜ Hψ ( Y, Y ′ ) ≤ ˜ Hϕ ( X, X ′ ) ≤ _ x ′ ∈ X ′ ϕ ( z, x ′ ) ≤ _ z ′ ∈ X ′ + Y ′ ( ϕ + ψ )( z, z ′ ) , and ( ¶ ) follows. (cid:3) The previous proof shows that, without the assumption k = ⊤ , one has that+ : G H × G H → G H is a V -functor, e.g. that ( G H, +) is a monoid in the Cartesiancategory V - CAT , but here we will continue to consider the monoidal structure of V - CAT . Theorem 6. If k = ⊤ in V , then the Hausdorff functor H : V - Cat → V - Cat induces a homomorphism H : ( G H, +) → ( G H, × ) of monoids in the monoidalcategory V - CAT .Proof.
Let us first show that the object-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 ( HX, HY ) forall V -categories X, Y . But for every V -module ϕ : X −→◦ Y and all A ⊆ X one has˜ Hϕ ( X, Y ) ≤ ˜ Hϕ ( A, Y ) ≤ _ B ⊆ Y ˜ Hϕ ( A, B ) , which implies ˜ Hϕ ( X, Y ) ≤ ˜ H ( ˜ Hϕ )( HX, HY )and then the desired inequality.In order to identify H as a homomorphism, 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. neutralin ( G H, × ). The bijective map+ : HX × HY → 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 all A, A ′ ⊆ X , B, B ′ ⊆ Y . With a = 1 ∗ X and b = 1 ∗ Y , in the notation of theproof of Proposition 5 one has _ z ′ ∈ A ′ + B ′ ( a + b )( x, z ′ ) = _ x ′ ∈ A ′ a ( x, x ′ ) N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 23 for all x ∈ A (since ( 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. (cid:3) Remarks . (1) The ( V - Cat )-isomorphism HX × HY ∼ = H ( X + Y )exhibited in the proof of Theorem 6 easily extends to the infinite case: Y i HX i ∼ = H ( X i X i ) . (2) Since there is no general concept of a (covariant!) functor transformingcoproducts into products, a more enlightening explanation for the formulajust encountered seems to be in order, as follows. Since V - Cat is an exten-sive category (see [6]), for every (small) family ( X i ) i ∈ I of V -categories thefunctor Σ : Y i V - Cat /X i → V - Cat / X i X i is an equivalence of categories. Now, the (isomorphism classes of a) commacategory V - Cat /X can be made into a (large) V -category when we definethe V -category structure c by c ( f, g ) = ^ x ∈ A _ y ∈ B ∗ X ( f ( x ) , g ( y )) = h X ( f ( A ) , g ( B )) , for all f : A → X , g : B → X in V - Cat . In this way the equivalenceΣ has become an isomorphism of V -categories, and since HX is just a V -subcategory of V - Cat /X , the ( V - Cat )-isomorphism of (1) is simply a re-striction of the isomorphism Σ: Q i V - Cat /X i P / / V - Cat / P i X i Q i HX i ?(cid:31) O O ∼ / / H ( P i X i ) ?(cid:31) O O Symmetrization A V -category X , or just its structure a = 1 ∗ X , is symmetric when a = a ◦ . Thisdefines the full subcategory V - Cat s of V - Cat which is coreflective: the coreflectorsends an arbitrary 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 all x, y ∈ X . By H s X = ( HX ) s = ( P X, h sX ) one can define a sublifting H s : V - Cat → V - Cat of the powerset functor which (like H ) preserves full fidelity, but which (unlike H ) fails to be a 2-functor. However itsrestriction H s : V - Cat s → V - Cat s is a 2-functor. Remarks . (1) H s X must not be confused with H ( X s ). For example, for V = and a set X provided with a separated (=antisymmetric) order, X s carriesthe discrete 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) Even after its restriction to V - Cat s there is no easy way of evaluating f H s ϕ ( A, B ) for a V -module ϕ : X −→◦ Y and A ⊆ X , B ⊆ Y , since the com-putation leading to the easy formula of Lemma 2 does not carry throughwhen H is replaced by H s .(3) The following addendum to Proposition 3 suggests how to overcome thedifficulty mentioned in (2) when trying to define a non-trivial symmetricGromov structure: V -category structures c on the disjoint union X ⊔ Y such that the V -categories X, Y become full V -subcategories correspondbijectively to pairs of V -modules ϕ : X −→◦ Y , ϕ ′ : Y −→◦ X with ϕ ′ · ϕ ≤ ∗ X , ϕ · ϕ ′ ≤ ∗ Y ;we write ϕ : X ◦ / / Y : ϕ ′◦ o o for such a pair. Under the hypotheses of Proposition 3 we can now write GK ( X, Y ) = _ ϕ : X ◦ / / Y : ϕ ′◦ o o ˜ Kϕ ( X, Y ) . Hence, for any sublifting K of P we put G s K ( X, Y ) := _ ϕ : X ◦ / / Y : ϕ ′◦ o o ˜ Kϕ ( X, Y ) ∧ ˜ Kϕ ′ ( Y, X )and obtain easily:
Corollary 3.
For any sublifting K of the powerset functor, G s K = ( G , G s K ) is a large symmetric V -category, and when K is a 2-functor preserving full fidelityof V -functors, then G s K ( X, Y ) = _ X֒ → Z ← ֓Y ∗ KZ ( X, Y ) ∧ ∗ KZ ( Y, X ) = _ X֒ → ( X ⊔ Y,c ) ← ֓Y ˜ Kc ( X, Y ) ∧ ˜ Kc ( Y, X ) for all V -categories X, Y .Proof.
Revisiting the proof of Proposition 2, we just note that ϕ : X ◦ / / Y : ϕ ′◦ o o , ψ : Y ◦ / / Z : ψ ′◦ o o implies ψ · ϕ : X ◦ / / Z : ϕ ′ · ψ ′◦ o o . A slight adaption of the computation given in Proposition 2 now shows that G s K is indeed a V -category structure on G = ob V - Cat / ∼ =. The given formulae follow asin the proof of Proposition 3. (cid:3) N THE CATEGORICAL MEANING OF HAUSDORFF AND GROMOV DISTANCES, I 25
Corollary 4. G s H ( X, Y ) = G ( H s )( X, Y ) , for all V -categories X, Y . Extending the notion of symmetry from V -categories to V -modules, we call a V -module ϕ : X −→◦ Y symmetric if X, Y are symmetric with ϕ ◦ · ϕ ≤ ∗ X and ϕ · ϕ ◦ ≤ ∗ Y ; we write ϕ : X ◦ / / Y : ϕ ◦◦ o o in this situation and define G s K ( X, Y ) := _ ϕ : X ◦ / / Y : ϕ ◦◦ o o ˜ Kϕ ( X, Y )for every sublifting K of P . Since symmetric V -modules compose, similarly toCorollary 3 one obtains: Corollary 5.
For any sublifting K of the powerset functor, G s K := (ob V - Cat s / ∼ = , G s K ) is a large V -category, and when K is a 2-functor preserving full fidelity of V -functors, then G s K ( X, Y ) = _ X֒ → Z ← ֓YZ symmetric ∗ KZ ( X, Y ) = _ X֒ → ( X ⊔ Y,c ) ← ֓Yc symmetric ˜ Kc ( X, Y ) for all symmetric V -categories X, Y . (cid:3) Remarks . (1) It is important to note that G s K is not symmetric, even when K = H . For V = P + , X a singleton and Y G s H ( X, Y ) = 0 while G s H ( Y, X ) = . Henceit is natural to consider 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 same example spaces of the Introduction 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 ) } = 12 . (2) When the symmetric V -categories X, Y are fully embedded into some V -category Z , they are also 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) Instead of the coreflector X 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 analogy to H s X and G s X , respectively. Corollary 3 remainsvalid when s is traded for sym and ∧ for ⊗ . Separation A V -category X , or just its structure a = 1 ∗ X , is separated when k ≤ a ( x, y ) ∧ a ( y, x ) implies x = y for all x, y ∈ X . It was shown in [10] (and it is easy toverify) that the separated V -categories form an epireflective subcategory of V - Cat :the image of X under the Yoneda functor y X : X → ˆ X serves as the reflector. Fur-thermore, there is a closure operator which describes separation of X equivalentlyby the closedness of the diagonal in X × X . (This description is not needed in whatfollows, but it further confirms the naturality of the concept.)In Remarks 2 we already presented a sublifting H ⇓ of the powerset functor,and it is easy to check that f H ⇓ ϕ ( A, B ) may be computed as ˜ Hϕ ( A, B ) in Lemma2, e.g. the two values coincide, because of the formula proved in Remarks 2(1).Furthermore, H ⇓ is like H a 2-functor which preserves full fidelity of V -functors.Hence, Proposition 3 is applicable to H ⇓ and may in fact be sharpened to: Corollary 6.
For all separated V -categories X, Y one has GH ( X, Y ) = GH ⇓ ( X, Y ) = _ X֒ → Z ← ֓YZ separated h Z ( X, Y ) = _ X֒ → ( X ⊔ Y,c ) ← ֓Yc separated ˜ Hc ( X, Y ) . Proof.
The structure c constructed from a V -module ϕ as in the proof of Proposition3 is separated. (cid:3) Remarks . (1) From Corollary 3 one obtains G s H ( X, Y ) = G s H ⇓ ( X, Y ) = _ X֒ → Z ← ֓Y h Z ( X, Y ) ∧ h Z ( Y, X )= _ X֒ → ( X ⊔ Y,c ) ← ֓Y ˜ Hc ( X, Y ) ∧ ˜ Hc ( Y, X ) . However, here it is not possible to restrict the last join to separated struc-tures c : consider the trivial case when V = and X, Y are singleton sets.(2) V -category structures c on X ⊔ Y that are both symmetric and separatedcorrespond bijectively to symmetric modules ϕ : X −→◦ Y with k ϕ ( x, y )for all x ∈ X , y ∈ Y , provided that X and Y are both symmetric andseparated. For V = , X, Y are necessarily discrete, and the only structure c is discrete as well.(3) The structure GH on G is not separated, even if we consider only isomor-phism classes of separated V -categories: for V = , the order on G given by GH is chaotic! Likewise when G is traded for G s . References [1] A. Akhvlediani,
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