Metric Realization in Persistence and Geometry

• Metric Realization is the process of constructing metric (or extended pseudo-metric) spaces from combinatorial, simplicial, or persistence data, with explicit control over distances.
• It employs adjunctions between persistence diagrams and ep-metric spaces via functors like Re and S, ensuring higher simplices contribute metric relations only through their 1-skeleton.
• Applications include Vietoris–Rips realizations, Wasserstein metric thickenings, inverse-limit models, and graph-coordinate representations, addressing both homotopy and embedding challenges.

Metric realization, in the literature considered here, denotes constructions that associate genuine metric or extended pseudo-metric spaces to combinatorial, simplicial, or persistence data, and, conversely, encode metric spaces by singular, inverse-limit, or coordinate models. A central instance is the functor $Re:sSet^{[0,\infty]}\to ep\text{-}Met$, which takes a persistence diagram in simplicial sets to an extended pseudo-metric space and admits a right adjoint singular functor $S$. Related developments realize simplicial thickenings as Wasserstein metric spaces, rectilinear Gromov–Hausdorff geodesics as Hausdorff geodesics in an ambient space, and certain spaces as inverse limits of metric graphs (Jardine, 2020, Adams et al., 2021, Ivanov et al., 2019, Cheeger et al., 2011).

1. Realization of persistence diagrams as ep-metric spaces

In the framework of "Metric spaces and homotopy types" (Jardine, 2020), one works with the functor category $sSet^{[0,\infty]}$, whose objects are diagrams

$Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$

and with the category $ep\text{-}Met$ of extended pseudo-metric spaces. An ep-metric space is a set $X$ equipped with a function

$d:X\times X\to [0,\infty]$

satisfying $d(x,x)=0$, $d(x,y)=d(y,x)$, and $d(x,z)\le d(x,y)+d(y,z)$. The construction is extended because distances may be infinite, and pseudo because one does not require $S$0.

For each $S$1 and each scale $S$2, the equilateral $S$3-simplex $S$4 in $S$5 has underlying set $S$6 and metric

$S$7

Given a persistence diagram $S$8, the realization is defined by the colimit

$S$9

where $sSet^{[0,\infty]}$0 is the translation category of simplices and $sSet^{[0,\infty]}$1 is the representable diagram which is $sSet^{[0,\infty]}$2 in degrees $sSet^{[0,\infty]}$3 and empty below $sSet^{[0,\infty]}$4.

The same paper gives an explicit metric description. The underlying set of $sSet^{[0,\infty]}$5 is the set of $sSet^{[0,\infty]}$6-simplices of $sSet^{[0,\infty]}$7. If two vertices $sSet^{[0,\infty]}$8 lie in distinct path-components of $sSet^{[0,\infty]}$9, then

$Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$0

If $Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$1 lie in the same component, one considers finite sequences of $Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$2-simplices

$Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$3

forming a polygonal path

$Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$4

sets

$Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$5

and defines

$Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$6

This recovers the usual weighted-edge metric on the $Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$7-skeleton, extended by $Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$8 between components.

2. Skeleton dependence and the adjunction $Y:[0,\infty]\to sSet,\qquad s\mapsto Y_s,$9

A key structural fact is that $ep\text{-}Met$0 depends only on the $ep\text{-}Met$1-skeleton of the persistence diagram (Jardine, 2020). If $ep\text{-}Met$2 denotes the diagram of $ep\text{-}Met$3-skeleta of $ep\text{-}Met$4, then the inclusion $ep\text{-}Met$5 induces an isomorphism

$ep\text{-}Met$6

in $ep\text{-}Met$7. The reason is that, in the colimit defining $ep\text{-}Met$8, every $ep\text{-}Met$9-simplex with $X$0 contributes no new metric relations beyond those coming from its $X$1-faces, because in $X$2 every pair of distinct vertices already has distance $X$3. A common misconception is therefore excluded: higher simplices affect homotopy-theoretic structure, but they do not add further metric relations to $X$4.

The right adjoint of $X$5 is the singular functor

$X$6

For an ep-metric space $X$7, it is defined by

$X$8

Equivalently, an $X$9-simplex of $d:X\times X\to [0,\infty]$0 is a bag of points $d:X\times X\to [0,\infty]$1 in $d:X\times X\to [0,\infty]$2 such that $d:X\times X\to [0,\infty]$3 for all $d:X\times X\to [0,\infty]$4. Faces and degeneracies are the usual ones.

The adjunction is expressed by natural bijections

$d:X\times X\to [0,\infty]$5

A non-expanding map $d:X\times X\to [0,\infty]$6 is uniquely determined by its composites

$d:X\times X\to [0,\infty]$7

for each simplex $d:X\times X\to [0,\infty]$8, and those composites exactly give a natural transformation $d:X\times X\to [0,\infty]$9. This places metric realization within a categorical duality between persistence diagrams and ep-metric spaces.

3. Vietoris–Rips realization and persistent homotopy type

For a finite ep-metric space $d(x,x)=0$0 equipped with some total order, the Vietoris–Rips diagram is

$d(x,x)=0$1

(Jardine, 2020). In this case the realization functor is exact in a strong sense: $d(x,x)=0$2 The underlying sets agree. If $d(x,x)=0$3 in $d(x,x)=0$4, then there is a single edge $d(x,x)=0$5 between $d(x,x)=0$6 and $d(x,x)=0$7, so $d(x,x)=0$8. Conversely, any polygonal path in $d(x,x)=0$9 corresponds to a chain in $d(x,y)=d(y,x)$0 whose total length is $d(x,y)=d(y,x)$1, so the infimum in $d(x,y)=d(y,x)$2 recovers $d(x,y)=d(y,x)$3.

The adjunction counit at $d(x,y)=d(y,x)$4 is a natural map

$d(x,y)=d(y,x)$5

which in simplicial degree $d(x,y)=d(y,x)$6 and scale $d(x,y)=d(y,x)$7 sends the ordered simplex $d(x,y)=d(y,x)$8 in $d(x,y)=d(y,x)$9 to the bag $d(x,z)\le d(x,y)+d(y,z)$0. Jardine’s Theorem 16 states that if $d(x,z)\le d(x,y)+d(y,z)$1 is totally ordered, then for each $d(x,z)\le d(x,y)+d(y,z)$2 the map

$d(x,z)\le d(x,y)+d(y,z)$3

is a weak homotopy equivalence of simplicial sets. More precisely, the induced map on nerves of non-degenerate simplices is a homotopy equivalence, and the inclusion of the subdivision is also a weak equivalence.

The significance is homotopical rather than metric: $d(x,z)\le d(x,y)+d(y,z)$4 and $d(x,z)\le d(x,y)+d(y,z)$5 have the same homotopy type for all $d(x,z)\le d(x,y)+d(y,z)$6. In particular, one may replace the usual Vietoris–Rips complex by the potentially infinite singular complex $d(x,z)\le d(x,y)+d(y,z)$7 without changing any persistent homotopy or (co)homology. This suggests that metric realization serves not only to recover an ep-metric space from a filtration, but also to compare standard and singular models of persistent topology.

4. Wasserstein metric realization of simplicial thickenings

A different realization theory appears in "Operations on Metric Thickenings" (Adams et al., 2021). Let $d(x,z)\le d(x,y)+d(y,z)$8 be a metric space. Denote by $d(x,z)\le d(x,y)+d(y,z)$9 the set of all finitely supported probability measures on $S$00, and by $S$01 the set of all Radon probability measures with finite $S$02th moment. The space $S$03 is equipped with the $S$04-Wasserstein metric

$S$05

where $S$06 is the set of couplings of $S$07 and $S$08. The space $S$09 becomes a metric subspace.

A simplicial metric thickening of $S$10 is any subspace $S$11 such that $S$12, $S$13, lands in $S$14, and whenever $S$15 and $S$16, then $S$17. The category $S$18 has objects $S$19, where $S$20 is a metric space, $S$21 is an abstract simplicial complex, and $S$22 is a bijection on vertex sets. A morphism $S$23 consists of a short map $S$24 and a simplicial map $S$25 satisfying $S$26 on vertices.

The metric realization functor

$S$27

is defined on objects by

$S$28

with the induced $S$29-Wasserstein metric, and on morphisms by push-forward $S$30. In this setting Vietoris–Rips and Čech thickenings are recovered as

$S$31

The categorical structure has strong homotopy consequences. If $S$32 and $S$33, then there is a natural homotopy equivalence

$S$34

and similarly for wedge sums under the hypotheses stated in the paper. For all $S$35 and $S$36,

$S$37

The paper emphasizes that this overcomes two classical defects of ordinary geometric realization of possibly non-locally-finite complexes: non-metrizability and the discontinuity of the inclusion $S$38.

5. Alternative realization frameworks in metric geometry

Metric realization also appears in several distinct but structurally related forms. In "Hausdorff Realization of Linear Geodesics of Gromov-Hausdorff Space" (Ivanov et al., 2019), a rectilinear Gromov–Hausdorff geodesic $S$39 arising from a closed optimal correspondence $S$40 is realized as a shortest path in an explicitly built ambient metric space. Writing $S$41, the metric

$S$42

has the property that each slice $S$43 is isometric to $S$44, and

$S$45

Thus the original rectilinear GH-geodesic becomes a genuine constant-speed geodesic in the Hausdorff metric.

In "Realization of metric spaces as inverse limits, and bilipschitz embedding in $S$46" (Cheeger et al., 2011), realization takes the form of an inverse-limit representation. If $S$47 is $S$48-Lipschitz and Lipschitz-light with constant $S$49, then for any integer $S$50 there is an admissible inverse system of directed metric graphs $S$51 with inverse limit $S$52, together with compatible $S$53-Lipschitz maps $S$54, such that the induced map

$S$55

is $S$56-bilipschitz and

$S$57

Moreover, if $S$58 is any admissible system, then $S$59 admits a $S$60-Lipschitz map into some $S$61 whose inverse-Lipschitz constant depends only on $S$62.

In "Metric representations by minimal graphs" (Franco-Sánchez et al., 5 Feb 2026), realization is combinatorial. For a resolving set $S$63, the metric representation of $S$64 is

$S$65

A finite set $S$66 is realizable if there exists a graph $S$67 and a resolving set $S$68 such that

$S$69

The canonical realization $S$70 has vertex-set $S$71 and edge-set

$S$72

and every realization is isomorphic to a spanning subgraph of $S$73. The paper distinguishes edge-minimal and minimum realizations, gives a necessary and sufficient condition for removing an edge while preserving all metric coordinates, proves that BMETREL is NP-complete, and characterizes the vector sets realizable by a tree.

Taken together, these constructions show that metric realization is not a single formalism but a recurring pattern: simplicial, categorical, GH-geodesic, inverse-limit, and graph-coordinate data can each be converted into a metric object with explicit control of distances, geodesics, or resolving coordinates.

6. Obstructions and regularity thresholds

Metric realization is also constrained by regularity and target-category obstructions. A particularly sharp example is given in "A two-dimensional $S$74 metric with no local $S$75 embedding in $S$76, following Pogorelov" (Holland, 2012). On the disk $S$77, the rotationally symmetric metric

$S$78

has $S$79 of class $S$80, and the paper proves that there need not exist a local isometric embedding of class $S$81

$S$82

At the same time, the same metric admits an elementary $S$83 realization as a surface of revolution, obtained by rotating the profile curve

$S$84

The nonexistence proof uses the Gauss curvature formula

$S$85

a developable-surface argument, the existence of arbitrarily short affine segments on the flat disk, a local graph representation $S$86, and a one-variable convexity lemma. The resulting contradiction shows that even a metric whose coefficients have two continuous derivatives and Lipschitz second derivatives may fail to admit a local isometric embedding of class $S$87. The paper states that the threshold regularity is "strictly above" $S$88.

This point is also the natural place to separate two notions that are often conflated. Metric realization in the sense of $S$89, Wasserstein thickenings, or graph-coordinate realizability is a constructive passage from discrete or simplicial data to a metric space. Isometric realization in $S$90 is instead an embedding problem for a prescribed Riemannian metric. The Pogorelov example shows that success in the former sense does not imply success in the latter. The same paper notes that Pogorelov’s result is somewhat controversial among the community of researchers that study isometric immersions, in part because of the lack of details in Pogorelov’s original paper; its purpose is therefore to provide the missing details while preserving the original construction.