Operations on Metric Thickenings
DDavid I. Spivak and Jamie Vicary (Eds.):Applied Category Theory 2020 (ACT2020)EPTCS 333, 2021, pp. 261–275, doi:10.4204/EPTCS.333.18
Operations on Metric Thickenings
Henry Adams Johnathan Bush Joshua Mirth
Colorado State UniversityColorado, USA { lastname } @math.colostate.edu Many simplicial complexes arising in practice have an associated metric space structure on the vertexset but not on the complex, e.g. the Vietoris–Rips complex in applied topology. We formalize aremedy by introducing a category of simplicial metric thickenings whose objects have a naturalrealization as metric spaces. The properties of this category allow us to prove that, for a large classof thickenings including Vietoris–Rips and ˇCech thickenings, the product of metric thickenings ishomotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.
Applied topology studies geometric complexes such as the Vietoris–Rips and ˇCech simplicial complexes.These are constructed out of metric spaces by combining nearby points into simplices. We observe thatproofs of statements related to the topology of Vietoris–Rips and ˇCech simplicial complexes often containa considerable amount of overlap, even between the different conventions within each case (for example, ≤ versus < ). We attempt to abstract away from the particularities of these constructions and considerinstead a type of simplicial metric thickening object. Along these lines, we give a natural categoricalsetting for so-called simplicial metric thickenings [3].In Sections 2 and 3, we provide motivation and briefly summarize related work. Then, in Section 4,we introduce the definition of our main objects of study: the category MetTh of simplicial metric thick-enings and the associated metric realization functor (cid:3) m from MetTh to the category of metric spaces.We define
MetTh as a particular instance of a comma category and prove that this definition satisfiescertain desirable properties, e.g. it possesses all finite products. We define simplicial metric thickeningsas the image of the metric realization of
MetTh . Particular examples of interest include the Vietoris–Ripsand ˇCech simplicial thickenings.In Section 5, we prove that certain (co)limits are preserved, up to homotopy equivalence, by thefunctors defined in Section 4. For example, we show that the metric realization functor factors overproducts and wedge sums. We also prove that the analogous (co)limits are preserved for the Vietoris–Rips and ˇCech simplicial thickenings.
Our motivation is twofold: first to give a general and categorical definition of simplicial metric thick-enings, which first appeared in [3], though primarily in the special case of the Vietoris–Rips metricthickenings. Second, to use this framework to give succinct proofs about the homotopy types of theseobjects while de-emphasizing the particular details of the Vietoris–Rips or ˇCech complex constructions.Let us first explain the reason to consider an alternative to the simplicial complex topology. Whilethe vertex set of a Vietoris–Rips or ˇCech complex is a metric space, the simplicial complex itself may not62
Operations on Metric Thickenings be. A simplicial complex is metrizable if and only if it is locally finite, meaning each vertex is containedin only a finite number of simplices, and a Vietoris–Rips complex (with positive scale parameter) isnot locally finite if it is not constructed from a discrete metric space. Similarly, the inclusion of ametric space, X , into its Vietoris–Rips or ˇCech complex is not continuous unless X is discrete, sincethe restriction of the simplicial complex topology to the vertex set is the discrete topology. Both of theseproblems are addressed by the Vietoris–Rips and ˇCech metric thickenings , which are metric spaces andfor which there is a canonical isometric embedding of the underlying metric space.As an example, let us consider in more detail the differences between the Vietoris–Rips simplicialcomplex and the Vietoris–Rips metric thickening at the level of objects and morphisms. Given a metricspace X , the Vietoris–Rips simplicial complex VR ( X ; r ) has as its simplicies all finite subsets σ ⊆ X ofdiameter at most r . We interpret this construction as an element of the image of a bifunctor VR ( (cid:3) ; (cid:3) ) with domain Met × [ , ∞ ) , where [ , ∞ ) is the poset ([ , ∞ ) , ≤ ) viewed as a category, and with codomain sCpx , the category of simplicial complexes and simplicial maps. There is then a geometric realizationfunctor from sCpx to the category of topological spaces. For a fixed metric space X , we have a functorfrom [ , ∞ ) to topological spaces, in particular, a morphism VR ( X ; r ) (cid:44) → VR ( X ; r (cid:48) ) whenever r ≤ r (cid:48) .As a simplicial complex, VR ( X ; 0 ) contains a vertex for each point of X and no higher-dimensionalsimplices. However, if X is not a discrete metric space, then VR ( X ; 0 ) and X may not even be homotopyequivalent because VR ( X ; 0 ) is the set X equipped with the discrete topology. Problems arise also for r >
0, when VR ( X ; r ) need not be metrizable—a simplicial complex is metrizable if and only if it islocally finite. In contrast, Vietoris–Rips metric thickenings are a functor VR m ( (cid:3) ; (cid:3) ) from Met × [ , ∞ ) to metric spaces, not just to topological spaces. In particular, VR m ( X ; 0 ) is isometric to X . Furthermore,given a 1-Lipschitz map X → Y , we obtain a natural transformation VR m ( X ; (cid:3) ) → VR m ( Y ; (cid:3) ) . So,VR m ( (cid:3) ; (cid:3) ) is indeed a bifunctor from Met × [ , ∞ ) to Met .There is a fair bit known about Vietoris–Rips complexes VR ( X ; r ) that does not immediately transferto the metric thickenings VR m ( X ; r ) . Some properties, such as the stability of persistent homology [10],are potentially difficult to transfer in a categorical fashion. Other properties, such as statements aboutproducts and wedge sums, do transfer over cleanly.Whereas proofs about homotopy types of Vietoris–Rips and ˇCech simplicial complexes often involvesimplicial collapses, the corresponding proofs for metric thickenings instead often involve deformationretractions not written as a sequence of simplicial collapses. We give two versions of this correspondencein Section 5, including explicit formulas proving that the Vietoris–Rips thickening of an L ∞ product(respectively wedge sum) of metric spaces deformation retracts onto the product (wedge sum) of theVietoris–Rips thickenings. Hence, thickenings behave nicely with respect to certain limits and colimits. This paper draws on three distinct bodies of work. The topology of Vietoris–Rips and ˇCech complexeshas been widely studied in the applied topology community [2, 4, 16, 17, 25, 37]. Major questions includedetermining the homotopy type of the Vietoris–Rips complex of a given space at all scale parameters r (see in particular [2] which determines all Vietoris–Rips complexes of the circle), and of determiningthe topology of the Vietoris–Rips complex of a product, wedge sum, or other gluing of spaces whoseindividual Vietoris–Rips complexes are known. Metric gluings were studied extensively in [4], andproducts in [8, 15]. Here we study similar questions, not about the Vietoris–Rips simplicial complex butthe Vietoris–Rips metric thickening. These latter objects were introduced in [3].A well-known construction is the metric of barycentric coordinates, which is a metrization of any .H. Adams, J.E. Bush & J.R. Mirth K , as explained in [7, Section 7A.5], and can be considered a functor B : sCpx → Met .Consider a real vector space V with basis K , the vertex set of K , equipped with an inner product (cid:104)− , −(cid:105) such that this basis is orthonormal. We can realize K as the set of all finite, convex, R -linear combi-nations of basis vectors (i.e. vertices) contained in some simplex. The inner product defines a metric, d b ( u , v ) = (cid:112) (cid:104) u − v , u − v (cid:105) , on V . The restriction of this metric to K is called the metric of barycentriccoordinates. Dowker proves in [12] that the identity map from a simplicial complex K with the simpli-cial complex topology to K with the metric of barycentric coordinates is a homotopy equivalence. A keydifference between the simplicial metric thickenings considered in this paper and the metric of barycen-tric coordinates is the following: with barycentric coordinates (as with the simplicial complex topology)the vertex set is equipped with the discrete topology, but in a simplicial metric thickening the vertex setneed not be discrete. Another functor from simplicial complexes to metric spaces is studied in [27, 28].This functor also produces a space with the same (weak) homotopy type as the geometric realization.Roughly, this construction is to take a simplicial complex K and consider the space of random variables X : Ω → K where Ω is some reference probability space and K denotes the vertex set of K . The space L ( Ω , K ) which metrizes K is the subset of random variables which give positive probability to all subsetsof K which correspond to simplices in K , and the metric is given by the measure (in Ω ) of the set onwhich two random variables differ. This construction also places the discrete topology on the vertex set K , and therefore typically disagrees with the homotopy type of the simplicial metric thickening.Finally, we draw some inspiration from the idea of a probability monad in applied category theory.A probability monad, or more specifically the Kantorovich monad [14, 30], is a way to put probabilitytheory on a categorical footing. A probability monad P is defined so that if X is a metric space, then PX is a collection of random elements in X . As the main data of the monad, there is an evaluation map PPX (cid:55)→ PX defined by averaging. Furthermore, an algebra of the probability monad, i.e. an evaluationmap PX (cid:55)→ X , is analogous to a Karcher or Fr´echet mean map as used in the proof of [3, Theorem 4.2]and [6, Theorem 4.6, Theorem 5.5]. Moreover, the Kantorovich monad of [14] places the Wassersteinmetric on the space of probability measures, as we do when defining simplicial metric thickenings. We begin by fixing some notation. Given a metric space X , let P X denote the set of all Radon probabilitymeasures on X with finite p -th moment. With the p -Wasserstein metric, P X is a metric space; for detailssee Section 4.3. There is a canonical inclusion δ : X → P X given by δ ( x ) = δ x . To avoid a proliferationof subscripts we will also write δ ( x ) . We will write ν (cid:28) µ to mean that ν is absolutely continuous withrespect to µ , that is, if whenever µ ( E ) = E ⊆ X , then ν ( E ) =
0. Let I X denotethe subspace of P X consisting of measures with finite support, i.e. those of the form µ = ∑ ni = λ i δ ( x i ) . Definition 4.1. A simplicial metric thickening of a metric space X is a subspace K of I X whichsatisfies:1. The image of δ : X → I X is contained in K , and2. If µ ∈ K and ν (cid:28) µ , then ν ∈ K . As a point of comparison, recall the definition of an abstract simplicial complex:
Definition 4.2. An abstract simplicial complex on a set V is a subset K of V consisting only of finitesets which satisfies1. The image of the map v (cid:55)→ { v } is in K, and2. If σ ∈ K and τ ⊆ σ , then τ ∈ K. Operations on Metric Thickenings
Example 4.3.
A motivating example of a simplicial thickening is the Vietoris–Rips simplicial metricthickening [3]. Recall the Vietoris–Rips complex, VR ( X ; r ) , as described in Section 2. By necessity ofthe construction, the vertices of VR ( X ; r ) have an associated metric, even though VR ( X ; r ) need not bemetrizable. The associated simplicial metric thickening, VR m ( X ; r ) , is the subset of P X containing allmeasures whose support set is a simplex in VR ( X ; r ) , and it is a metric space.We will frequently return to this example. In particular, the homotopy type of the Vietoris–Ripscomplex of various spaces is widely studied [2, 3, 9, 10, 16, 17, 35, 36, 37]. By formulating a categoryof simplicial metric thickenings which includes Vietoris–Rips thickenings, we are able to compute thehomotopy type of Vietoris–Rips thickenings of spaces constructed from limit and colimit operations.There are several reasonable choices of morphisms between simplicial metric thickenings. Since theyare metric spaces, any map of metric spaces could be allowed (see Section 4.2 for several choices of mapsof metric spaces). Alternatively, one could define a morphism between simplicial metric thickenings K and L of metric spaces X and Y , respectively, to be a function f : X → Y such that the pushforward f : K → P Y has its image contained in L . In Sections 4.1 and 4.2 we construct a description of acategory which has as objects the simplicial metric thickenings of Definition 4.1 and for which this latterdefinition of morphisms arises naturally. We work with the standard notions of category theory; for further details, we refer the reader to [31] (forexample). We often abuse notation and write c ∈ C when c is an object of the category C . Definition 4.4.
Given functors S : A → C and T : B → C , the comma category ( S ↓ T ) has as objects all triples ( a , b , φ ) where a ∈ A , b ∈ B , and φ : Sa → T b, and asmorphisms all pairs ( f , g ) with f ∈ Hom A ( a , a (cid:48) ) and g ∈ Hom B ( b , b (cid:48) ) , such that thefollowing diagram commutes: SaSa (cid:48) T bT b (cid:48) φ S f T g φ (cid:48) We introduce the following subcategory of a comma category.
Definition 4.5.
The restricted comma category [ S ↓ T ] is the full subcategory defined to contain allobjects ( a , b , φ ) ∈ ( S ↓ T ) such that φ is an isomorphism. For an arbitrary comma category, the order of the source functor S and target functor T is important: ( S ↓ T ) and ( T ↓ S ) are not equivalent as categories in general. However, restricted comma categories areless particular. Proposition 4.6.
The categories [ S ↓ T ] and [ T ↓ S ] are isomorphic. The proof of Proposition 4.6 is omitted. Our main theorems in Section 5 are about various types oflimits and colimits. As we explain below, restricted comma categories inherit these structures from theirsource and target categories.Observe that any comma category ( S ↓ T ) has two functors P A : ( S ↓ T ) → A and P B : ( S ↓ T ) → B , the domain and codomain functors. These are given by sending a triple ( a , b , φ ) to a and to b , respectively,and by sending a morphism ( f , g ) to f and to g , respectively. We will denote the functors P A : [ S ↓ T ] → A and P B : [ S ↓ T ] → B with the same symbols. Lemma 4.7.
Fix categories A , B , and C and functors S : A → C and T : B → C . For some small indexcategory J , suppose that A and B admit colimits under J -shaped diagrams and that S preserves colimitsunder J -shaped diagrams. Then ( S ↓ T ) admits colimits under J -shaped diagrams. .H. Adams, J.E. Bush & J.R. Mirth Dually, if A and B admit limits over J -shaped diagrams and T preserves limits over J -shaped dia-grams, then ( S ↓ T ) admits limits over J -shaped diagrams.Proof. We will prove only the case for colimits; the case for limits follows by dualizing the proof.Let D : J → ( S ↓ T ) be a diagram in the comma category, and denote the objects in its image by ( a j , b j , φ j ) for j ∈ J . Then P A D : J → A and P B D : J → B are J -shaped diagrams in A and B , and sohave colimits (cid:96) a and (cid:96) b . There is a natural transformation Φ = ( φ j ) j ∈ J : SP A D = ⇒ T P B D . Observethat SP A D : J → C is a diagram in C with colimit S (cid:96) a because S preserves colimits. Let Z : P B D = ⇒ (cid:96) B denote the cocone natural transformation. Then T Z Φ : SP A D = ⇒ T (cid:96) B is a cocone over SP A D , so thereexists a unique morphism ψ : S (cid:96) A → T (cid:96) B (see Figure 1).The colimit of D is ( (cid:96) A , (cid:96) B , ψ ) . Indeed, suppose that ( a , b , χ ) is a cocone over D . Then there areunique morphisms f : (cid:96) A → a and f : (cid:96) B → b because composition with P A or P B gives diagrams in A and B . The morphism ( f , f ) ∈ Hom ( S ↓ T ) (( (cid:96) A , (cid:96) B , φ ) , ( a , b , χ )) is well-defined because everything insight commutes. Hence, ( S ↓ T ) admits colimits under J -shaped diagrams. Sa i · · · Sa j S (cid:96) a T b i · · · T b j T (cid:96) b φ i φ j ψ Figure 1: There exists a map, ψ , because T (cid:96) B is a cone over SP A D . Corollary 4.8.
With the setup of Section 5, suppose that the image of D : J → ( S ↓ T ) is contained in thesubcategory [ S ↓ T ] . Then the limit over (respectively, colimit under) D is contained in [ S ↓ T ] .Proof. In the special case that D : J → [ S ↓ T ] is a diagram in the restricted comma category, the naturaltransformations Φ has an inverse, Φ − = ( φ − j ) j ∈ J . It follows that S (cid:96) A is a colimit under T P B D and T (cid:96) B is a colimit under SP A D . Therefore, there are unique morphisms ψ : S (cid:96) A → T (cid:96) B and ξ : T (cid:96) B → S (cid:96) A andthese are necessarily isomorphisms. Hence, [ S ↓ T ] admits colimits under J -shaped diagrams. Lemma 4.9.
Let P A and P B be the domain and codomain functors from ( S ↓ T ) to A and B , respectively.If T has a left adjoint then so does P A , and if S has a right adjoint so does P B .Proof. To begin, we assume that T has a left adjoint L , with counit ε : LT = ⇒ id B and unit η : id C = ⇒ T L (see [31, Section 4.2], for example). Define (cid:101) L : A → ( S ↓ T ) by A (cid:51) a (cid:55)→ ( a , LSa , η Sa ) ∈ ( S ↓ T ) Hom A ( a , a ) (cid:51) f (cid:55)→ ( f , LS f ) ∈ Hom ( S ↓ T ) ( (cid:101) La , (cid:101) La ) . Operations on Metric Thickenings ( S ↓ T ) A BC
TSP A P B L (cid:101) L (cid:101) L (cid:101) LP A (cid:101) L (cid:101) L (cid:101) L (cid:101) η id (cid:101) L (cid:101) ε (cid:101) L P A P A (cid:101) LP A P A (cid:101) η P A id P A P A (cid:101) ε Figure 2: (Left) The setup of Lemma 4.9. (Middle and Right) Triangle identities for an adjunction.We claim that (cid:101) L is left adjoint to P A . Observe that P A (cid:101) L = id A so there is trivially a unit (cid:101) η : id A = ⇒ P A (cid:101) L . We need to construct a counit (cid:101) ε : (cid:101) LP A = ⇒ id ( S ↓ T ) . Define (cid:101) ε ( a , b , φ ) = ( id a , ε b ◦ L φ ) , and observe thatthe triangle identities in Figure 2 (middle and right) are satisfied for this definition of counit.In particular, the triangle identity of Figure 2 (middle) is satisfied because (cid:101) ε (cid:101) La ◦ (cid:101) L (cid:101) η a = (cid:101) ε (cid:101) La ◦ id (cid:101) La = (cid:101) ε ( a , LSa , η Sa ) = ( id a , ε LSa ◦ L η Sa ) = ( id a , id LSa ) = id (cid:101) La for all a ∈ A , and the triangle identity Figure 2 (right) is satisfied because P A (cid:101) ε c ◦ (cid:101) η P A c = P A (cid:101) ε c ◦ id P A c = P A ( id a , ε b ◦ L φ ) = id a = id P A c for all c = ( a , b , φ ) ∈ ( S ↓ T ) .A similar argument shows that if S has a right adjoint R then P B has a right adjoint (cid:101) R . Corollary 4.10.
Let P A and P B be the domain and codomain functors from [ S ↓ T ] to A and B , respectively.If S has a left or right adjoint, then so does P B , and likewise if T has a left or right adjoint, so does P A .Proof. Apply Lemma 4.9 to see that if S has a right adjoint, then so does P B , and that if T has a leftadjoint, then so does P A . The case of when S has a left adjoint or T has a right adjoint follows after firstapplying Proposition 4.6. To formalize simplicial metric thickenings as comma categories, we first recall the definitions of thecategories of simplicial complexes and of metric spaces.
Definition 4.11.
Let K and L be simplicial complexes with vertex sets K and L . A simplicial map is afunction f : K → L such that if σ is a simplex of K, then f ( σ ) is a simplex of L. The category of simplicial complexes, sCpx , has abstract simplicial complexes as objects and simpli-cial maps as morphisms. This category admits finite products and coproducts. The categorical productof simplicial complexes K and L , denoted K ∏ L , is the simplicial complex such that ( σ , τ ) ∈ K ∏ L is asimplex whenever σ ∈ K and τ ∈ L [22, Definition 4.25]. The coproduct, denoted K ∏ L , is the disjointunion simplicial complex. Definition 4.12.
Let X and Y be metric spaces and k ∈ [ , + ∞ ) . A function f : X → Y is k -Lipschitz ifd ( f ( x ) , f ( x (cid:48) )) ≤ kd ( x , x (cid:48) ) for all x , x (cid:48) ∈ X . Functions which are -Lipschitz may be called short . Lipschitz functions are, of course, continuous. We define the category of metric spaces,
Met , to havemetric spaces as objects and short maps as morphisms. While this is a standard definition (it is the sameused in [14], for example), there are alternative definitions in the literature, where either the morphismsare less-restricted, or the axioms of a metric space are relaxed [23]. In particular, the morphisms may be .H. Adams, J.E. Bush & J.R. Mirth k -Lipschitz for some k ∈ [ , ∞ ) , or simply continuous maps. The latter isthe structure of the category of metric spaces as a full subcategory of Top . Many of our constructions donot depend on the choice of morphisms for
Met , but our default choice in this paper is short maps.The metric space axioms may also be relaxed when defining a category of metric spaces. Recallthat the classical definition of a metric space is a set X equipped with a function d ( · , · ) : X × X → [ , ∞ ) such that d ( x , y ) = x = y , d ( x , y ) = d ( y , x ) for all x , y ∈ X , and d ( x , z ) ≤ d ( x , y ) + d ( y , z ) for all x , y , z ∈ X . Allowing d ( x , y ) = ∞ gives an extended metric space . Allowing d ( x , y ) = x (cid:54) = y gives a pseudo-metric space . Allowing d ( x , y ) (cid:54) = d ( y , x ) is a quasi-metric space . Combining allof the three above relaxations gives Lawvere metric spaces, or categories enriched in the monoidal poset ([ , + ∞ ] , ≤ , +) .We will make use of classical metric spaces and of extended pseudo-metric spaces, denoting thecategory of the latter by pMet . Of course, Met is a full subcategory of pMet .The category
Met has finite products. If X and Y are metric spaces, the product X × Y is the carte-sian product of the underlying sets with the supremum norm: d (( x , y ) , ( x (cid:48) , y (cid:48) )) = max { d ( x , x (cid:48) ) , d ( y , y (cid:48) ) } .Coproducts do not exist in Met ; however, colimits under certain other diagrams do, including the wedgesum discussed in Section 5.2.One advantage of pMet is the existence of arbitrary products and coproducts. The product is definedusing the supremum metric, and the coproduct X ∏ Y is the set X (cid:116) Y with d ( x , y ) = + ∞ for x ∈ X and y ∈ Y (all other distances are unchanged). The necessity of working in pMet for arbitrary products isshown by the following example (see [26, Chapter 2, Example 1.9] for a formal proof that arbitraryproducts may not exist in Met ). Consider the space X = R N (that is, sequences of real numbers) with thesupremum norm. The distance between x = ( , , . . . , , . . . ) and y = ( , , , . . . , n , . . . ) is then d ( x , y ) = sup n ∈ N n = ∞ . All the other axioms of the metric are still satisfied by supremums taken over infinite sets,however.Note that both the categories of metric spaces and simplicial complexes possess canonical functorsto Set . For metric spaces, the functor U is given by forgetting the metric d , U : Met (cid:51) ( X , d ) (cid:55)→ X ∈ Set f : ( X , d X ) → ( Y , d Y ) (cid:55)→ f : X → Y For abstract simplicial complexes, the functor (cid:3) is given by forgetting the subset structure, (cid:3) : sCpx (cid:51) K (cid:55)→ K ∈ Set f : K → L (cid:55)→ f | K : K → L Here K and L are the vertex sets of the simplicial complexes K and L . We will often not refer to U and (cid:3) explicitly and instead write X or K to refer to the underlying sets. Definition 4.13.
The category
MetTh of simplicial metric thickenings is the re-stricted comma category [ U ↓ (cid:3) ] . Explicitly, objects are triples ( X , K , φ ) , in whichX is a metric space, K is an abstract simplicial complex, and φ : K → X is an iso-morphism of sets, and a morphism between ( X , K , φ ) and ( Y , L , ψ ) is a pair of shortmaps ( f : X → Y , g : K → L ) such that the following diagram commutes in Set : XK YL φ fg | K ψ Note that the source category of U can be either Met or pMet , to distinguish we use MetTh and pMetTh . Next, we establish some basic properties of the category of simplicial metric thickenings.68
Operations on Metric Thickenings
Proposition 4.14.
The domain and codomain functors P pMet : pMetTh → pMet and P sCpx : pMetTh → sCpx both have left and right adjoints. In addition, the functor P Met also defines a functor
MetTh → Met with left and right adjoints.Proof.
As per Corollary 4.10, we only need to show that U and (cid:3) have adjoints. Starting with (cid:3) , theright adjoint is the complete simplicial complex functor, C , and the left adjoint is the trivial complexfunctor, T .Let D r : Set → pMet be the functor giving every set the discrete metric where all distances are equalto r . The right adjoint of U is D and the left adjoint is D ∞ . These are not defined for Met , and so P sCpx has adjoints only in pMetTh , and not in MetTh .Note that sCpx and
Met can both be embedded into
MetTh . Choosing some r , the functor D r : sCpx → MetTh is a full and faithful embedding, and the functors T : Met → MetTh and C : Met → MetTh arefull and faithful embeddings.
Proposition 4.15. If pMet and sCpx each admit (co)limits over small diagrams of shape J , then so does pMetTh . If Met and sCpx each posses limits over small diagrams of shape J , then so does MetTh . Inparticular, pMetTh admits finite products and coproducts, and
MetTh admits finite products.Proof.
As described in Proposition 4.14, (cid:3) and U both have left and right adjoints. Therefore, both arecontinuous and cocontinuous functors, i.e., they preserve small limits and colimits. By Corollary 4.8, [ U ↓ (cid:3) ] has limits of any small diagram for which limits exist in both Met and sCpx . Here we show that every object of pMetTh can be realized as a space satisfying Definition 4.1. We callthis realization the metric realization of the simplicial metric thickening. It was first introduced in [3]and is related to [14]. Much like the convention for geometric realizations of a simplicial complexes, wewill often not distinguish between an object of
MetTh and its metric realization.As a point of comparison, there is a functor | (cid:3) | : sCpx → Top that takes a simplicial complex K toa topological space | K | called the geometric realization. While simplicial thickenings could be givena topology using | (cid:3) | and factoring through P sCpx , the metric realization functor provides a more directtopological realization with better properties due to the metric structure. As described in [3], the metricthickening of a simplicial thickening ( X , K , φ ) in which K is locally-finite is always homeomorphic tothe geometric realization of the simplicial complex | K | . However, geometric realizations of non-locally-finite complexes are non-metrizable, so the metric thickening topology is necessarily different.To define the metric realization, we need a certain number of measure-theoretic definitions. If X is ametric space, we will consider it a measurable space with its Borel σ -algebra. Given a point x ∈ X , let δ ( x ) denote the delta distribution with mass one centered at the point x . By a probability measure on X we mean a Radon measure µ such that µ ( X ) =
1. We will furthermore assume that probability measureshave finite moments, meaning that for any fixed x (cid:48) ∈ X and p ∈ [ , + ∞ ) , we have (cid:82) X d X ( x , x (cid:48) ) p d µ ( x ) < ∞ .Note that any measure with finite support and total mass one is a probability Radon measure with finitemoments. Denote the set of all probability Radon measures with finite moments on X by P X . Recallalso that the support of a measure µ is the (closed) setsupp ( µ ) = { x ∈ X | µ ( A ) > A (cid:51) x } . The technical restrictions on the measures in P X are necessary for P X to be a metric space underthe Wasserstein (also Kantorovich or earth-movers) distance. .H. Adams, J.E. Bush & J.R. Mirth Definition 4.16.
Let X be a metric space and let µ , ν be probability measures on X . Let Γ ( µ , ν ) be theset of all measures π on X × X such that π ( X , A ) = ν ( A ) and π ( B , X ) = µ ( B ) for all measurable sets Aand B (that is, all measures whose marginals are µ and ν ). The p -Wasserstein distance between µ and ν is d ( µ , ν ) : = inf π ∈ Γ ( µ , ν ) (cid:18) (cid:90) X × X d X ( x , y ) p d π (cid:19) / p . For more details on the Wasserstein distance, including the fact that it defines a metric on P X andthat all choices of p are topologically equivalent, see [3, 13, 20, 21, 33, 34].We now have the requisite machinery to define the metric realization of a simplicial metric thicken-ing. Definition 4.17.
The metric realization functor (cid:3) m : pMetTh → pMet is specified by the followingdata: • For each simplicial thickening K = ( X , K , φ ) in pMetTh , let K m be the sub-metric space of P Xof all probability measures µ such that φ ( supp ( µ )) = σ for some σ ∈ K. • For each morphism ( f , g ) : ( X , K , φ ) → ( Y , L , ψ ) , let ( f , g ) m be the morphism taking µ = ∑ ni = λ i δ ( x i ) to f m ( µ ) = ∑ ni = λ i δ ( f ( x i )) . Note that this also restricts to a functor (cid:3) m : MetTh → Met . There is no difficulty in allowingpseudo-metric spaces here, even though many references only treat classical metric spaces. If X containssome point x (cid:48) with d ( x (cid:48) , x ) = ∞ for some x (and hence all y within finite distance of that x ), then nomeasure with x , x (cid:48) ∈ supp ( µ ) is in P X due to the finite moments condition. Pseudo-metric spaces alsohave a natural topology and a well-defined Borel σ -algebra, so P X is defined for such spaces.The objects here are precisely those described by Definition 4.1. Indeed, for finitely-supported mea-sures, we have ν (cid:28) µ if and only if supp ( ν ) ⊆ supp ( µ ) . Therefore the morphisms are precisely functions f : X → Y between metric spaces such that the pushforward map f : K m → P Y has its image containedin L m . This holds for any of the variants of categories of metric spaces described in Section 4.2, thoughin the following we always take Met or pMet with short maps as morphisms.As described earlier, the Vietoris–Rips complex provides a natural example of the construction ofsimplicial thickenings. The above definitions allow us to describe the Vietoris–Rips complex as a functor: Definition 4.18.
Let r ∈ [ , + ∞ ] . The Vietoris–Rips functor VR ( (cid:3) ; r ) : Met → MetTh is defined by VR ( (cid:3) ; r ) : Met (cid:51) X (cid:55)→ ( X , VR ( X ; r ) , id ) f : X → Y (cid:55)→ ( f , f ) This is well-defined because f is a short map and therefore sends any simplex σ to a set of points withno larger diameter. The Vietoris–Rips simplicial thickening is the composition of functors VR m ( (cid:3) ; r ) .A related construction is the ˇCech complex functor. In a metric space X , we let B r ( x ) denote the ballof radius r centered at the point x ∈ X . Definition 4.19.
Let r ∈ [ , + ∞ ] and let X be a set. The ˇCech complex , ˇCech ( X ; r ) , has a simplex forevery finite subset σ ⊆ X such that ∩ x ∈ σ B r ( x ) (cid:54) = /0 . The ˇCech functor ˇCech ( (cid:3) ; r ) : Met → MetTh isdefined by ˇCech ( (cid:3) ; r ) : Met (cid:51) X (cid:55)→ ( X , ˇCech ( X ; r ) , id ) f : X → Y (cid:55)→ ( f , f ) Again, the ˇCech simplicial thickening is the composition ˇCech m ( (cid:3) ; r ) . We will study both of theseconstructions further in Section 5.70 Operations on Metric Thickenings
Vietoris–Rips and ˇCech simplicial complexes preserve certain homotopy properties under products andwedge sums. Indeed, the case of ( L ∞ ) products is given in [2, Proposition 10.2], [15], [25] and the caseof wedge sums is given in [4, 5, 9, 24].In this section we give categorical proofs for metric thickenings. We have seen that if Met and sCpx have (co)limits of a certain shape, then so does
MetTh . We now prove that certain (co)limits arepreserved by the metric thickening functors (cid:3) m , VR m ( (cid:3) ; r ) , and ˇCech m ( (cid:3) ; r ) , at least up to homotopytype. We begin with the product operation. The deformation retraction we construct corresponds to the mapsending a measure on a product space to the product measure of its corresponding marginals.We use × to denote the product in Met and sCpx , and ∏ for the product in MetTh . Since productsexist in both
Met and sCpx , they exist in
MetTh by Proposition 4.15. Explicitly, the product of M =( X , K , φ ) and N = ( Y , L , ψ ) is M ∏ N = ( X × Y , K × L , φ × ψ ) . Proposition 5.1.
For any simplicial metric thickenings M and N, the metric realization factors over theproduct up to homotopy: M m × N m (cid:39) ( M ∏ N ) m . Proof.
Let M = ( X , K , φ ) and N = ( Y , L , ψ ) . Elements of M m are finitely-supported measures ofthe form µ = ∑ i λ i δ ( x i ) with x i ∈ X and φ ( supp ( µ )) ∈ K . Likewise elements of N m have the form ν = ∑ j η j δ ( y j ) with y j ∈ Y and ψ ( supp ( ν )) ∈ L . Thus elements of M m × N m are pairs ( µ , ν ) =( ∑ i λ i δ ( x i ) , ∑ j η j δ ( y j )) with φ ( supp ( µ )) × ψ ( supp ( ν )) ∈ K × L . On the other hand, elements of ( M ∏ N ) m are measures on X × Y of the form ∑ i , j α i , j δ (( x i , y j )) .With this in mind, there is is an obvious injection ι : M m × N m (cid:44) → ( M ∏ N ) m via (cid:32) ∑ i λ i δ ( x i ) , ∑ j η j δ ( y j ) (cid:33) (cid:55)→ ∑ i , j λ i η j δ (( x i , y j )) . Concretely, ι sends a pair of measures on X and Y to their product measure on X × Y . There is also asurjection ρ : ( M ∏ N ) m (cid:16) M m × N m given by taking the marginals of the joint distribution: ∑ i , j α i , j δ (( x i , y j )) (cid:55)→ (cid:32) ∑ i (cid:18) ∑ j α i , j (cid:19) δ ( x i ) , ∑ j (cid:18) ∑ i α i , j (cid:19) δ ( y j ) (cid:33) . We now show that ι and ρ are homotopy inverses. Certainly ρ ◦ ι = id by construction. Note that thecomposition ι ◦ ρ gives the map ∑ i , j α i , j δ (( x i , y j )) (cid:55)→ ∑ i , j (cid:32) ∑ i α i , j (cid:33) (cid:32) ∑ j α i , j (cid:33) δ (( x i , y j )) . This is homotopic to the identity on ( M ∏ N ) m via the straight-line homotopy H : ( M ∏ N ) m × I → ( M ∏ N ) m where H ( t , µ ) = t id ( µ ) + ( − t ) ι ◦ ρ ( µ ) . This is clearly well-defined as a map to P ( X × Y ) . To see that the image of H is in ( M ∏ N ) m , note that supp ( ι ◦ ρ ( µ )) ⊆ supp ( µ ) , so the entirehomotopy takes place within a simplex of K × L . It then follows from [3, Lemma 3.9] that homotopy H is continuous. .H. Adams, J.E. Bush & J.R. Mirth Corollary 5.2.
For any metric spaces X and Y , the product operation factors through the metric Vietoris–Rips and ˇCech thickenings up to homotopy: VR m ( X × Y ; r ) (cid:39) VR m ( X ; r ) × VR m ( Y ; r ) ˇCech ( X × Y ; r ) (cid:39) ˇCech ( X ; r ) × ˇCech ( Y ; r ) . Proof.
As simplicial complexes, we have an isomorphism VR ( X × Y ; r ) ∼ = VR ( X ; r ) ∏ VR ( Y ; r ) sincewith the L ∞ metric, a subset of X × Y has diameter equal to the maximum of the diameters of its co-ordinate projections. Similarly, we have an isomorphism ˇCech ( X × Y ; r ) ∼ = ˇCech ( X ; r ) ∏ ˇCech ( Y ; r ) of ˇCech simplicial complexes since a collection of L ∞ balls intersect if and only if their projectionsonto both factors intersect. Thus VR m ( X × Y ; r ) ∼ = ( VR ( X ; r ) ∏ VR ( Y ; r )) m and ˇCech m ( X × Y ; r ) ∼ = (cid:0) ˇCech ( X ; r ) ∏ ˇCech ( Y ; r ) (cid:1) m . The result then follows from Proposition 5.1. Proposition 5.3.
The metric thickening functors (cid:3) m , VR m ( (cid:3) ; r ) , and ˇCech m ( (cid:3) ; r ) all preserve coprod-ucts.Proof. We are working in the category pMet of pseudo-metric spaces, where coproducts exist. Recallthe coproduct X ∏ Y has d ( x , y ) = + ∞ for x ∈ X and y ∈ Y . Hence the simplicial metric thickenings (cid:3) m ,VR m ( (cid:3) ; r ) , and ˇCech m ( (cid:3) ; r ) of a coproduct are simply the coproducts of the thickenings. Though Proposition 5.3 is somewhat uninteresting, another colimit operation to consider is the wedgesum.
Definition 5.4.
Let • be the terminal object in a category C . Given A , B ∈ C , • A : • → A, and • B : • → B, the wedge sum of A and B, denoted A ∨ B, is thepushout of • A and • B : • A BA ∨ B • A • B ι A ι B Proposition 5.5.
Wedge sums exist in
Met , sCpx , and MetTh .Proof.
The description of the wedge sum in each category is essentially the same. The terminal object in
Met is the metric space with a single point. The wedge sum X ∨ Y is the metric space X (cid:116) Y / ( • X ∼ • Y ) ,that is, X and Y are “glued together” at the points • X and • Y . We will refer to this common basepointin X ∨ Y as • . The metric on X ∨ Y is given by d ( x , y ) = d ( x , • ) + d ( • , y ) for x ∈ X and y ∈ Y , whiledistances within X and Y are unchanged. One can check that, with this metric, X ∨ Y satisfies theappropriate universal property.The terminal object in sCpx is the simplicial complex with a single vertex. The wedge sum K ∨ L isthe simplicial complex K (cid:116) L / ( • K ∼ • L ) , and again we refer to the common basepoint as • .Since wedge sums exist in both Met and sCpx , they exist in
MetTh by Proposition 4.15. The wedgesum of M = ( X , K , φ ) and N = ( Y , L , ψ ) is M ∨ N = ( X ∨ Y , K ∨ L , φ ∨ ψ ) . Remark 5.6.
For any simplicial metric thickenings M and N , the metric realization factors over thewedge sum. Indeed, we have M m ∨ N m = ( M ∨ N ) m . However, if F : Met → MetTh , it is too muchto expect that F ( X ∨ Y ) ∼ = F ( X ) ∨ F ( Y ) . This fails, for example, if F is the Vietoris–Rips functor; seeFigures 3 and 4. Therefore proving that the metric thickening behaves well with respect to wedge sumsis more delicate than the product case.72 Operations on Metric Thickenings Figure 3: VR ( X ; r ) ∨ VR ( Y ; r ) Figure 4: VR ( X ∨ Y ; r ) Proposition 5.7.
Let M = ( X , K , φ ) and N = ( Y , L , ψ ) be simplicial thickenings. Suppose the simplicialthickening V = ( X ∨ Y , S , φ ) has the property that S ⊇ K ∨ L, and if σ ∈ S is a subset of neither φ ( X ) nor ψ ( Y ) , then σ ∪ • is also a simplex in S. Then V m (cid:39) ( M ∨ N ) m .Proof. Elements of both V m and ( M ∨ N ) m have the form εδ ( • ) + ∑ i λ i δ ( x i ) + ∑ j η j δ ( y j ) where ε + λ + η =
1. (Here we assume x i ∈ X and y j ∈ Y , and define λ = ∑ i λ i and η = ∑ j η j .)Further, elements of ( M ∨ N ) m must satisfy λ = η =
0. Since S ⊇ K ∨ L , there is an inclusion ι : ( M ∨ N ) m (cid:44) → V m .Define ρ : V m (cid:16) ( M ∨ N ) m by εδ ( • ) + ∑ i λ i δ ( x i ) + ∑ j η j δ ( y j ) (cid:55)→ (cid:40) ( η + ε ) δ ( • ) + (cid:0) − ηλ (cid:1) ∑ i λ i δ ( x i ) if λ ≥ η ( λ + ε ) δ ( • ) + (cid:16) − λη (cid:17) ∑ j η j δ ( y j ) if η ≥ λ , setting ηλ = λ = λη = η =
0. To see that ρ is continuous, note that the twopiecewise formulas agree when λ = η (in which case the image of ρ is • ). By construction the image of ρ is in ( M ∨ V ) m , and ρ is in fact a deformation retract, so ρ ◦ ι = id.To complete the proof, ι ◦ ρ is homotopic to the identity via H ( t , µ ) = t id ( µ ) + ( − t ) ι ◦ ρ ( µ ) . Twocases are required to show that the image of H is indeed V m . If supp ( µ ) ⊆ X or supp ( µ ) ⊆ Y , thensupp ( ι ◦ ρ ( µ )) = supp ( µ ) . Otherwise supp ( ι ◦ ρ ( µ )) = supp ( µ ) ∪ • . Regardless, ( φ ∨ ψ )( supp ( µ ) ∪ supp ( ι ◦ ρ ( µ ))) is a simplex in S by assumption. It then follows from [3, Lemma 3.9] that the homotopy H is continuous. Corollary 5.8.
For any metric spaces X and Y , the wedge sum factors through the metric Vietoris–Ripsand ˇCech thickenings up to homotopy: VR m ( X ∨ Y ; r ) (cid:39) VR m ( X ; r ) ∨ VR m ( Y ; r ) ˇCech m ( X ∨ Y ; r ) (cid:39) ˇCech m ( X ; r ) ∨ ˇCech m ( Y ; r ) . Proof.
The Vietoris–Rips case follows since VR ( X ∨ Y ; r ) ⊇ VR ( X ; r ) ∨ VR ( Y ; r ) , and since if σ ∈ VR ( X ∨ Y ; r ) is not a subset of either X or Y , then σ ∪ • ∈ VR ( X ∨ Y ; r ) . The ˇCech case is analogous.We remark that in Corollary 5.8, the same proof (the homotopy equivalence from Proposition 5.7)works equally well whether X and Y are finite or infinite. By contrast, proofs of analogous statementsfor Vietoris–Rips and ˇCech simplicial complexes either don’t apply to the infinite setting [24], or alter-natively need to treat the infinite setting as a separate case [4]. .H. Adams, J.E. Bush & J.R. Mirth We give a categorical definition for certain constructions arising in applications of topological data analy-sis, namely, metric thickenings of a simplicial complex. The utility of this approach is seen in the conciseproofs and organizational schema afforded by the language of category theory. In particular, we intro-duce two equivalent definitions of the category
MetTh of simplicial metric thickenings and prove thatthis category possesses a number of desirable properties, such as the existence of forgetful functors withleft and right adjoints to both the category of metric spaces and the category of simplicial complexes.We define metric realizations of the simplicial metric thickenings in
MetTh as images of the metric real-ization functor (cid:3) m . We specialize to Vietoris–Rips and ˇCech metric thickenings by precomposing withappropriate functors from Met to MetTh . Furthermore, we prove that products and wedge sums factorthrough the resulting metric Vietoris–Rips and ˇCech thickenings.We end with some open questions.1. Is the stability of persistent homology afforded by Vietoris–Rips and ˇCech simplicial complexes [10]also shared by simplicial metric thickenings? See [3, Conjecture 6.14].2. Is VR m < ( X ; r ) homotopy equivalent to VR < ( X ; r ) for any metric space X and scale r >
0, andsimilarly for ˇCech thickenings? Here the subscript < means that a finite set is included as asimplex if its diameter is strictly less than r .3. If one instead allows measures of infinite support, how much does this affect the homotopy typeof a simplicial metric thickening? Acknowledgements
We would like to thank Alex McCleary and Amit Patel for their support of the Category Theory Lab atColorado State University.
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