---
title: Interleaving Distance of Sheaves
url: https://www.emergentmind.com/topics/interleaving-distance-of-sheaves
type: topic
---

# Interleaving Distance of 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 [2108.13018].

## 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 \(k\) is a field. In the abstract formulation of Petit–Schapira, the basic input is a thickening kernel, namely a monoidal presheaf on \((\mathbb R_{\ge 0},+)\) with values in the monoidal category of derived kernels on \(X\). If \(\mathfrak K=(\mathfrak K_a)_{a\ge 0}\) is such a family, it satisfies \(\mathfrak K_0\simeq k_{\Delta_X}\), \(\mathfrak K_a\circ \mathfrak K_b\simeq \mathfrak K_{a+b}\), and transition maps \(\rho_{b,a}:\mathfrak K_b\to \mathfrak K_a\) for \(a\le b\). A central structural result is that once a monoidal presheaf is defined on an interval containing \(0\), it extends uniquely to all of \(\mathbb R_{\ge 0}\); the restriction functor from global thickenings to local data is an equivalence of categories [2006.13150].

For metric spaces, the thickening is realized by the diagonal. If \((X,d_X)\) is a good metric space with constant \(\alpha_X>0\), then for all \(a,b\ge 0\) with \(a+b\le \alpha_X\), three properties are required: the intersections \(B_a(x_1)\cap B_b(x_2)\) are contractible or empty; the projections \(q_i:\Delta_a\to X\) are proper; and \(\Delta_a\circ \Delta_b\simeq \Delta_{a+b}\), where \(\Delta_a=\{(x,y)\mid d_X(x,y)\le a\}\). Under these hypotheses, \(a\mapsto k_{\Delta_a}\) defines a monoidal presheaf on \([0,\alpha_X]\), hence a unique thickening kernel on all of \(\mathbb R_{\ge 0}\) [2006.13150].

In the real-analytic setting used for the nondegeneracy theorem, one assumes that \(X\) 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 \(k_{\Delta_a}\) are invertible up to shift for \(0<a<\alpha_X\), and the induced endofunctors
\[
T_a:D^b(k_X)\to D^b(k_X),\qquad F\mapsto F*k_{\Delta_a}
\]
behave like translation functors [2108.13018].

## 2. Definition of interleavings and the pseudo-metric

Given a thickening kernel \(\mathfrak K\), one says that \(F,G\in D^b(k_X)\) are \(a\)-isomorphic if there exist morphisms
\[
f:\mathfrak K_a\circ F\to G,\qquad g:\mathfrak K_a\circ G\to F
\]
such that the two usual \(2a\)-shift diagrams commute. The associated interleaving distance is
\[
d_{\mathfrak K}(F,G)=\inf\{\,a\ge 0\mid F,G\text{ are }a\text{-isomorphic}\,\}\in[0,\infty].
\]
This distance is a pseudo-metric: symmetry and the triangle inequality follow from the monoidal identities for the thickening kernels [2006.13150].

In the metric-diagonal formulation, for \(\varepsilon\ge 0\) one writes
\[
T_\varepsilon(F)=F*k_{\Delta_\varepsilon}.
\]
An \(\varepsilon\)-interleaving between \(F\) and \(G\) is a pair of morphisms
\[
\phi:F\to T_\varepsilon(G),\qquad \psi:G\to T_\varepsilon(F)
\]
such that the composites \(T_\varepsilon(\phi)\circ\psi\) and \(T_\varepsilon(\psi)\circ\phi\) coincide with the canonical unit morphisms \(G\to T_{2\varepsilon}(G)\) and \(F\to T_{2\varepsilon}(F)\). One then sets
\[
d(F,G)=\inf\{\,\varepsilon\ge 0\mid \text{there exists an }\varepsilon\text{-interleaving between }F\text{ and }G\,\}.
\]
Again, \(d\) is immediately a pseudo-metric on objects of \(D^b(k_X)\) [2108.13018].

On \(\mathbb R\) and more generally on normed vector spaces, the same definition is usually written in convolution form. For \(F,G\in D^b(k_V)\), let \(K_r=k_{B(0,r)}\) for \(r\ge 0\), and define
\[
F\star G=Rs_!(F\boxtimes G).
\]
Then an \(\varepsilon\)-interleaving consists of morphisms
\[
f:K_\varepsilon\star F\to G,\qquad g:K_\varepsilon\star G\to F
\]
satisfying the standard commutative squares with \(K_{2\varepsilon}\to K_0\). The induced distance
\[
d_C(F,G)=\inf\{\varepsilon\ge 0\mid F\text{ and }G\text{ are }\varepsilon\text{-interleaved}\}
\]
is the convolution distance of Kashiwara–Schapira and its derived variants [1805.09694].

The central subtlety is that these constructions yield pseudo-distances a priori. The statement \(d(F,G)=0\) need not, on formal grounds alone, imply \(F\simeq G\). 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 \(X\) is a real analytic manifold with a good distance and the mild invertibility assumptions hold, then for any \(F,G\in D^b_c(k_X)\),
\[
d(F,G)=0 \quad\Longrightarrow\quad F\simeq G.
\]
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 [2108.13018].

The proof proceeds through a stronger intermediate notion, called \(a\)-isomorphism. In kernel form, \(F\) and \(G\) are \(a\)-isomorphic if there exist
\[
u_a:F*k_{\Delta_a}\to G,\qquad v_a:G*k_{\Delta_a}\to F
\]
such that the convolution composites agree with the canonical units. Under invertibility of the kernels \(k_{\Delta_a}\), this is equivalent to the \(\varepsilon\)-interleaving formulation with \(\varepsilon=a\) [2108.13018].

The key technical input is a rigidity property of the Hom-spaces \(\operatorname{Hom}(F*k_{\Delta_b},G)\) as \(b\) varies. For constructible \(F\) and \(G\), the assignment
\[
b\longmapsto \operatorname{Hom}(F*k_{\Delta_b},G)
\]
is identified with the \(0\)-th cohomology of a constructible sheaf on \(\mathbb R\), supported in \([0,\alpha_X)\). In particular, these Hom-spaces are locally constant on open intervals. A similar statement holds for \(\operatorname{Hom}(G*k_{\Delta_b},F)\). This local constancy makes it possible to lower a \(b\)-isomorphism to an \(a\)-isomorphism for slightly smaller \(a<b\), and then to pass to the limit \(a\to 0\) [2108.13018].

A corollary is that the assignment \(F\mapsto d(F,-)\) gives a faithful embedding of the isomorphism classes of constructible sheaves into a space of \(\mathbb R\)-valued functions equipped with the supremum norm. In the \(\gamma\)-sheaf setting, the same nondegeneracy holds on \(\gamma\)-constructible sheaves with compact support. In particular, the constructible setting excludes the pathological possibility that two nonisomorphic objects lie at zero interleaving distance [2108.13018].

## 4. \(\gamma\)-sheaves, observable categories, and persistence modules

A major specialization arises when \(V\) is a finite-dimensional real vector space and \(\gamma\subset V\) is a closed, convex, proper cone with nonempty interior. Endowing \(V\) with the \(\gamma\)-topology produces the category of \(\gamma\)-sheaves, denoted \(\operatorname{Mod}(k_{V_\gamma})\) or \(D(k_{V_\gamma})\) in the derived setting. For each \(v\in V\), translation \(x\mapsto x-v\) induces an exact endofunctor \(T_{v*}\), and for \(w\le v\) there is a canonical morphism \(X_{w,v}:T_{w*}\Rightarrow T_{v*}\). Two objects \(F,G\in D(k_{V_\gamma})\) are \(v\)-interleaved if there exist maps
\[
f:T_{v*}F\to G,\qquad g:T_{v*}G\to F
\]
satisfying the evident compatibility squares. Fixing any \(v\in \operatorname{int}\gamma\), one defines
\[
d_\gamma(F,G)=\inf\{c\ge 0\mid F\text{ and }G\text{ are }(c\cdot v)\text{-interleaved}\},
\]
and the resulting pseudo-metric does not depend on the choice of \(v\) [1902.09933].

This setting is linked to ordinary persistence modules by the Alexandrov topology \(V_a\) induced by the preorder \(x\le y\iff y-x\in\gamma\). The subcategory of ephemeral modules consists of those objects whose sheafification to \(V_\gamma\) vanishes; in one parameter, this is equivalent to saying that all structure maps \(M_s\to M_t\) are zero whenever \(s<t\). The quotient by ephemeral modules satisfies
\[
\operatorname{Mod}(k_{V_a})/\operatorname{Eph}\simeq \operatorname{Mod}(k_{V_\gamma}),
\]
and in one parameter this is the observable category [1902.09933].

The quotient functor \(B_*:D(k_{V_a})\to D(k_{V_\gamma})\) is exact and essentially surjective, with fully faithful right adjoint, and it preserves interleaving distance:
\[
d_a(F,G)=d_\gamma(B_*F,B_*G).
\]
Thus passage to \(\gamma\)-sheaves does not alter the interleaving geometry. Moreover, if \(F,G\in D(k_{V_\gamma})\) have \(\gamma\)-proper supports, then after equipping \(V\) with the gauge norm whose unit ball is
\[
B_\gamma=(v+\gamma)\cap(-v+\gamma),
\]
the \(\gamma\)-interleaving distance and the convolution distance coincide exactly:
\[
d_\gamma(F,G)=d_C(F,G).
\]
In the special case \(X=\mathbb R^n\) with a closed convex cone \(\gamma\), this recovers the usual interleaving distance of multidimensional persistence modules, and on \(\gamma\)-constructible sheaves with compact support the zero-distance rigidity theorem applies as well [1902.09933].

## 5. The one-dimensional derived theory and graded barcodes

On the real line, the theory becomes explicitly computable. In \(D^b_{\mathbb Rc}(k_\mathbb R)\), one uses the kernels
\[
K_\varepsilon=
\begin{cases}
k_{[-\varepsilon,\varepsilon]} & \varepsilon\ge 0,\\
k_{(-\varepsilon,\varepsilon)}[1] & \varepsilon<0,
\end{cases}
\]
and convolution
\[
F\star G=Rs_!(F\boxtimes G).
\]
The resulting convolution distance \(d_C\) 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 \(B(F)\). The main theorem is the derived isometry theorem:
\[
d_C(F,G)=d_B\bigl(B(F),B(G)\bigr),
\]
where \(d_B\) is the bottleneck distance on graded barcodes [1805.09694].

The proof uses an explicit computation of all morphism spaces
\[
R\!\operatorname{Hom}_{D^b_{\mathbb Rc}(k_\mathbb R)}\bigl(k_I[k],k_J[\ell]\bigr)
\]
for intervals \(I,J\) and degrees \(k,\ell\). 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 \(d_C=d_B\) is established, closedness follows: if \(d_C(F,G)\le \varepsilon\), then \(F\) and \(G\) are already \(\varepsilon\)-interleaved [1805.09694].

For indecomposable interval sheaves, the distance can be written explicitly. For example,
\[
d_C(k_{[a,b]},k_{[c,d]})=\max\{|a-c|,\ |b-d|\},
\]
and similarly
\[
d_C(k_{[a,b)},k_{[c,d)})=\max\{|a-c|,\ |b-d|\}.
\]
For open intervals,
\[
d_C(k_{(a,b)},k_{(c,d)})=\frac12\max\bigl\{|(a+b)-(c+d)|,\ |b-a-(d-c)|\bigr\}.
\]
These formulas exhibit the direct parallel between sheaf convolution and barcode matching in dimension one [1805.09694].

## 6. Relative, localized, and microlocal variants

The sheaf-theoretic notion of interleaving also admits relative formulations. If \(f:P\to Q\) is an order-preserving map and \(Q\) carries a superlinear family of translations \(T_\varepsilon\), then for a \(P\)-module \(M\) one defines the relative shift
\[
M_P^\varepsilon=f^*T_\varepsilon^*f_*M,
\]
and two \(P\)-modules are \(\varepsilon\)-interleaved relative to \(f\) if they admit the corresponding maps into relative shifts with the usual pentagon compatibilities. The induced distance
\[
d_I^f(M,N)=\inf\{\varepsilon\ge 0\mid M,N\text{ are }\varepsilon\text{-interleaved relative to }f\}
\]
satisfies an isometry theorem:
\[
d_I^f(M,N)=d_I(f_*M,f_*N).
\]
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 [2004.14286].

A different extension appears in microlocal sheaf theory. On localized derived categories such as
\[
D^b(\mathbb R^n,\dot T^*\mathbb R^n)
\quad\text{and}\quad
D^b(S^{n-1}\times\mathbb R,T^{*,+}),
\]
one defines a convolution distance \(d_{\mathrm{conv}}\) by thickening in all directions of \(\mathbb R^n\) and an interleaving distance \(d_{\mathrm{int}}\) by thickening only in the \(\mathbb R\)-direction on \(S^{n-1}\times\mathbb R\). The sheaf-theoretic Radon transform
\[
\mathcal R:D^b(\mathbb R^n,\dot T^*\mathbb R^n)\xrightarrow{\simeq}D^b(S^{n-1}\times\mathbb R,T^{*,+})
\]
intertwines the thickening functors and yields an isometry
\[
d_{\mathrm{conv}}(F,G)=d_{\mathrm{int}}(\mathcal R(F),\mathcal R(G)).
\]
This converts multi-directional convolution on \(\mathbb R^n\) into one-directional thickening after Radon transform [2506.12046].

At a broader categorical level, interleavings can be formulated as extension problems of functors with common codomain. In that framework, sheaves on \(\operatorname{Open}(X)\) 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 [1707.06288].

## 7. Stability, completeness, and structural limitations

The thickening-kernel formalism was designed to support stability statements. If \(K_1,K_2\in D^b(k_{Y\times X})\) are kernels and \(Y\) is a good metric space, then for every \(F\in D^b(k_X)\),
\[
\operatorname{dist}_Y(K_1\circ F,K_2\circ F)\le \operatorname{dist}_{Y\times X/X}(K_1,K_2).
\]
More generally, a \(\delta\)-Lipschitz kernel \(K\) induces a \(\delta\)-Lipschitz map on the corresponding sheaf categories:
\[
\operatorname{dist}_Y(K\circ F_1,K\circ F_2)\le \delta\,\operatorname{dist}_X(F_1,F_2).
\]
In particular, if \(f:X\to Y\) is a \(\delta\)-Lipschitz map, then its graph kernel is \(\delta\)-Lipschitz and
\[
\operatorname{dist}_Y(Rf_!F_1,Rf_!F_2)\le \delta\,\operatorname{dist}_X(F_1,F_2).
\]
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 [2006.13150].

Completeness is more delicate. In the Tamarkin category
\[
\mathcal D(X)=\{\,F\in D(\mathbf k_{X\times\mathbb R_t})\mid \operatorname{SS}(F)\cap\{\tau\le 0\}=\emptyset\,\},
\]
the interleaving pseudo-distance is induced by the kernel
\[
\mathfrak K_c=\mathbf k_{\Delta_X\times\{|t_1-t_2|\le c\}},
\]
and on \(\mathcal D(X)\) this becomes equivalent to translation in the \(t\)-variable. If a sequence \((F_n)\) satisfies
\[
d_{\mathfrak K}(F_n,F_{n+1})\le a_n,\qquad \sum_{n=0}^\infty a_n<\infty,
\]
then there exists \(F_\infty\) such that \((F_n,F_\infty)\) is \((2a_{\ge n},24a_{\ge n})\)-isomorphic for every \(n\), where \(a_{\ge n}=\sum_{k\ge n}a_k\). Consequently, every Cauchy sequence in \(\mathcal D(X)\) has a limit, and \(\mathcal D(X)\) is complete for the pseudo-metric \(d_{\mathcal D(X)}\) [2201.02598].

The theory also has sharp limitations. Through the sheaf-function correspondence, constructible sheaves determine constructible functions via local Euler characteristic. If \(\delta\) is any pseudo-metric on constructible functions that is dominated by the convolution distance \(d_C\), then \(\delta\) 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 [2207.06335].

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 \(\gamma\)-sheaf form, it becomes computable on \(\mathbb R\) via graded barcodes, it extends to localized and relative settings, and on constructible sheaves it avoids zero-distance pathologies by satisfying genuine nondegeneracy [2108.13018].

Source: https://www.emergentmind.com/topics/interleaving-distance-of-sheaves