Interleaving Distance of Sheaves

• Interleaving distance of sheaves is a pseudo-metric on derived categories defined by controlled thickenings, generalizing persistence modules.
• It relies on convolution with thickened diagonal kernels in D⁽ᵇ⁾(kₓ) and utilizes a monoidal presheaf framework to produce well-behaved interleavings.
• The theory connects to graded barcodes, γ-sheaves, and microlocal methods, offering stability results and nondegeneracy for constructible sheaves.

The interleaving distance of sheaves is a pseudo-distance on a derived category of sheaves obtained by comparing objects after controlled thickening. In the metric formulation, one works on $D^b(k_X)$ for a real analytic manifold $X$ endowed with a good distance, defines $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$ by convolution with the constant sheaf on the $\varepsilon$-thickened diagonal, and sets the distance to be the infimum of $\varepsilon$ for which two objects are $\varepsilon$-interleaved. This framework generalizes earlier constructions for persistence modules and for sheaves on $\mathbb R^n$, and on subanalytic constructible sheaves with compact support—or more generally, constructible sheaves up to infinity—the pseudo-distance becomes a genuine distance because distance zero forces isomorphism (Petit et al., 2021).

1. Formal framework

A standard ambient category is the bounded derived category $D^b(k_X)$ of sheaves of $k$-vector spaces on a space $X$, where $X$0 is a field. In the abstract formulation of Petit–Schapira, the basic input is a thickening kernel, namely a monoidal presheaf on $X$1 with values in the monoidal category of derived kernels on $X$2. If $X$3 is such a family, it satisfies $X$4, $X$5, and transition maps $X$6 for $X$7. A central structural result is that once a monoidal presheaf is defined on an interval containing $X$8, it extends uniquely to all of $X$9; the restriction functor from global thickenings to local data is an equivalence of categories (Petit et al., 2020).

For metric spaces, the thickening is realized by the diagonal. If $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$0 is a good metric space with constant $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$1, then for all $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$2 with $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$3, three properties are required: the intersections $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$4 are contractible or empty; the projections $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$5 are proper; and $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$6, where $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$7. Under these hypotheses, $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$8 defines a monoidal presheaf on $T_\varepsilon(F)=F*k_{\Delta_\varepsilon}$9, hence a unique thickening kernel on all of $\varepsilon$0 (Petit et al., 2020).

In the real-analytic setting used for the nondegeneracy theorem, one assumes that $\varepsilon$1 is a real analytic manifold endowed with a good distance in the sense of Petit–Schapira–Waas. Under mild additional local topological convexity hypotheses, the kernels $\varepsilon$2 are invertible up to shift for $\varepsilon$3, and the induced endofunctors

$\varepsilon$4

behave like translation functors (Petit et al., 2021).

2. Definition of interleavings and the pseudo-metric

Given a thickening kernel $\varepsilon$5, one says that $\varepsilon$6 are $\varepsilon$7-isomorphic if there exist morphisms

$\varepsilon$8

such that the two usual $\varepsilon$9-shift diagrams commute. The associated interleaving distance is

$\varepsilon$0

This distance is a pseudo-metric: symmetry and the triangle inequality follow from the monoidal identities for the thickening kernels (Petit et al., 2020).

In the metric-diagonal formulation, for $\varepsilon$1 one writes

$\varepsilon$2

An $\varepsilon$3-interleaving between $\varepsilon$4 and $\varepsilon$5 is a pair of morphisms

$\varepsilon$6

such that the composites $\varepsilon$7 and $\varepsilon$8 coincide with the canonical unit morphisms $\varepsilon$9 and $\varepsilon$0. One then sets

$\varepsilon$1

Again, $\varepsilon$2 is immediately a pseudo-metric on objects of $\varepsilon$3 (Petit et al., 2021).

On $\varepsilon$4 and more generally on normed vector spaces, the same definition is usually written in convolution form. For $\varepsilon$5, let $\varepsilon$6 for $\varepsilon$7, and define

$\varepsilon$8

Then an $\varepsilon$9-interleaving consists of morphisms

$\mathbb R^n$0

satisfying the standard commutative squares with $\mathbb R^n$1. The induced distance

$\mathbb R^n$2

is the convolution distance of Kashiwara–Schapira and its derived variants (Berkouk et al., 2018).

The central subtlety is that these constructions yield pseudo-distances a priori. The statement $\mathbb R^n$3 need not, on formal grounds alone, imply $\mathbb R^n$4. The nondegeneracy problem is therefore structural rather than terminological.

3. Zero-distance rigidity on constructible sheaves

The main rigidity theorem states that on the constructible subcategory the pseudo-distance is nondegenerate. More precisely, if $\mathbb R^n$5 is a real analytic manifold with a good distance and the mild invertibility assumptions hold, then for any $\mathbb R^n$6,

$\mathbb R^n$7

Thus, on subanalytic constructible sheaves with compact support—and more generally on constructible sheaves up to infinity—the convolution pseudo-distance is a genuine distance (Petit et al., 2021).

The proof proceeds through a stronger intermediate notion, called $\mathbb R^n$8-isomorphism. In kernel form, $\mathbb R^n$9 and $D^b(k_X)$0 are $D^b(k_X)$1-isomorphic if there exist

$D^b(k_X)$2

such that the convolution composites agree with the canonical units. Under invertibility of the kernels $D^b(k_X)$3, this is equivalent to the $D^b(k_X)$4-interleaving formulation with $D^b(k_X)$5 (Petit et al., 2021).

The key technical input is a rigidity property of the Hom-spaces $D^b(k_X)$6 as $D^b(k_X)$7 varies. For constructible $D^b(k_X)$8 and $D^b(k_X)$9, the assignment

$k$0

is identified with the $k$1-th cohomology of a constructible sheaf on $k$2, supported in $k$3. In particular, these Hom-spaces are locally constant on open intervals. A similar statement holds for $k$4. This local constancy makes it possible to lower a $k$5-isomorphism to an $k$6-isomorphism for slightly smaller $k$7, and then to pass to the limit $k$8 (Petit et al., 2021).

A corollary is that the assignment $k$9 gives a faithful embedding of the isomorphism classes of constructible sheaves into a space of $X$0-valued functions equipped with the supremum norm. In the $X$1-sheaf setting, the same nondegeneracy holds on $X$2-constructible sheaves with compact support. In particular, the constructible setting excludes the pathological possibility that two nonisomorphic objects lie at zero interleaving distance (Petit et al., 2021).

4. $X$3-sheaves, observable categories, and persistence modules

A major specialization arises when $X$4 is a finite-dimensional real vector space and $X$5 is a closed, convex, proper cone with nonempty interior. Endowing $X$6 with the $X$7-topology produces the category of $X$8-sheaves, denoted $X$9 or $X$00 in the derived setting. For each $X$01, translation $X$02 induces an exact endofunctor $X$03, and for $X$04 there is a canonical morphism $X$05. Two objects $X$06 are $X$07-interleaved if there exist maps

$X$08

satisfying the evident compatibility squares. Fixing any $X$09, one defines

$X$10

and the resulting pseudo-metric does not depend on the choice of $X$11 (Berkouk et al., 2019).

This setting is linked to ordinary persistence modules by the Alexandrov topology $X$12 induced by the preorder $X$13. The subcategory of ephemeral modules consists of those objects whose sheafification to $X$14 vanishes; in one parameter, this is equivalent to saying that all structure maps $X$15 are zero whenever $X$16. The quotient by ephemeral modules satisfies

$X$17

and in one parameter this is the observable category (Berkouk et al., 2019).

The quotient functor $X$18 is exact and essentially surjective, with fully faithful right adjoint, and it preserves interleaving distance: $X$19 Thus passage to $X$20-sheaves does not alter the interleaving geometry. Moreover, if $X$21 have $X$22-proper supports, then after equipping $X$23 with the gauge norm whose unit ball is

$X$24

the $X$25-interleaving distance and the convolution distance coincide exactly: $X$26 In the special case $X$27 with a closed convex cone $X$28, this recovers the usual interleaving distance of multidimensional persistence modules, and on $X$29-constructible sheaves with compact support the zero-distance rigidity theorem applies as well (Berkouk et al., 2019).

5. The one-dimensional derived theory and graded barcodes

On the real line, the theory becomes explicitly computable. In $X$30, one uses the kernels

$X$31

and convolution

$X$32

The resulting convolution distance $X$33 is the derived-sheaf analogue of the classical one-parameter interleaving distance. Every object splits noncanonically as a direct sum of shifts of constant sheaves on intervals, and the associated multiset of interval-degree pairs is its graded barcode $X$34. The main theorem is the derived isometry theorem: $X$35 where $X$36 is the bottleneck distance on graded barcodes (Berkouk et al., 2018).

The proof uses an explicit computation of all morphism spaces

$X$37

for intervals $X$38 and degrees $X$39. It then decomposes barcodes into central, left, and right parts. The left and right pieces are sent by fully faithful functors to ordinary one-parameter persistence modules, where the classical isometry theorem applies. The central part is handled by a combinatorial trigonalization argument together with Hall’s marriage theorem. Once the equality $X$40 is established, closedness follows: if $X$41, then $X$42 and $X$43 are already $X$44-interleaved (Berkouk et al., 2018).

For indecomposable interval sheaves, the distance can be written explicitly. For example,

$X$45

and similarly

$X$46

For open intervals,

$X$47

These formulas exhibit the direct parallel between sheaf convolution and barcode matching in dimension one (Berkouk et al., 2018).

6. Relative, localized, and microlocal variants

The sheaf-theoretic notion of interleaving also admits relative formulations. If $X$48 is an order-preserving map and $X$49 carries a superlinear family of translations $X$50, then for a $X$51-module $X$52 one defines the relative shift

$X$53

and two $X$54-modules are $X$55-interleaved relative to $X$56 if they admit the corresponding maps into relative shifts with the usual pentagon compatibilities. The induced distance

$X$57

satisfies an isometry theorem: $X$58 All constructions dualize from cosheaves to sheaves when the target category has limits, and this yields a theory of interleavings and pixelizations for sheaves on down-set lattices and cell-posets (Botnan et al., 2020).

A different extension appears in microlocal sheaf theory. On localized derived categories such as

$X$59

one defines a convolution distance $X$60 by thickening in all directions of $X$61 and an interleaving distance $X$62 by thickening only in the $X$63-direction on $X$64. The sheaf-theoretic Radon transform

$X$65

intertwines the thickening functors and yields an isometry

$X$66

This converts multi-directional convolution on $X$67 into one-directional thickening after Radon transform (Takiwaki, 26 May 2025).

At a broader categorical level, interleavings can be formulated as extension problems of functors with common codomain. In that framework, sheaves on $X$68 are treated as functors on a weighted category, and the interleaving distance is defined as the infimum of the weights of common extensions. This places sheaf interleavings inside categorical analogues of Hausdorff and Gromov–Hausdorff distance (Bubenik et al., 2017).

7. Stability, completeness, and structural limitations

The thickening-kernel formalism was designed to support stability statements. If $X$69 are kernels and $X$70 is a good metric space, then for every $X$71,

$X$72

More generally, a $X$73-Lipschitz kernel $X$74 induces a $X$75-Lipschitz map on the corresponding sheaf categories: $X$76 In particular, if $X$77 is a $X$78-Lipschitz map, then its graph kernel is $X$79-Lipschitz and

$X$80

For complete Riemannian manifolds with positive convexity radius, the metric thickening of the diagonal and the thickening induced by the geodesic-flow Hamiltonian coincide, so the two resulting interleaving distances agree (Petit et al., 2020).

Completeness is more delicate. In the Tamarkin category

$X$81

the interleaving pseudo-distance is induced by the kernel

$X$82

and on $X$83 this becomes equivalent to translation in the $X$84-variable. If a sequence $X$85 satisfies

$X$86

then there exists $X$87 such that $X$88 is $X$89-isomorphic for every $X$90, where $X$91. Consequently, every Cauchy sequence in $X$92 has a limit, and $X$93 is complete for the pseudo-metric $X$94 (Asano et al., 2022).

The theory also has sharp limitations. Through the sheaf-function correspondence, constructible sheaves determine constructible functions via local Euler characteristic. If $X$95 is any pseudo-metric on constructible functions that is dominated by the convolution distance $X$96, then $X$97 vanishes whenever two compactly supported constructible functions have the same Euler integral. One consequence stated for topological data analysis is that there cannot exist non-trivial additive invariants of persistence modules that are continuous for the interleaving distance (Berkouk, 2022).

These results delineate the present shape of the subject. Interleaving distance for sheaves is simultaneously a metric, categorical, and microlocal notion: it is built from thickening kernels, it recovers persistence-theoretic interleavings in $X$98-sheaf form, it becomes computable on $X$99 via graded barcodes, it extends to localized and relative settings, and on constructible sheaves it avoids zero-distance pathologies by satisfying genuine nondegeneracy (Petit et al., 2021).