Sheaf Convolution: Structures & Applications

• Sheaf convolution is a framework that integrates sheaf-theoretic data via group laws, kernel transforms, and Laplacian diffusion.
• It underpins constructions like monoidal products in derived categories, convolution algebras with sheaf coefficients, and interleaving metrics in persistence.
• Its applications span representation theory, noncommutative geometry, topological data analysis, and geometric deep learning, offering versatile tools for advanced analytics.

Sheaf convolution denotes a family of operations that combine sheaf-theoretic data by transporting an external tensor product, or an analogous coefficient system, along an ambient composition law such as vector-space addition, group multiplication, groupoid composition, or a sheaf Laplacian. In the literature, the term therefore does not refer to a single invariant construction but to several structurally related ones: monoidal products on derived categories of sheaves, convolution algebras with sheaf coefficients, integral transforms defined by kernels, convolution distances on sheaf categories, and graph- or manifold-based message passing in sheaf neural networks (Bezrukavnikov et al., 2022, Gonçalves et al., 2019, Hu, 4 Nov 2025, Peng et al., 22 Apr 2026).

1. Foundational patterns and terminological scope

A first recurrent model is the pushforward-along-addition formula. On a real vector space $E$, or on the Lie algebra $\mathfrak{gl}_n$, convolution is defined from the addition map $s$ or $a$ by

$$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$

This produces a symmetric monoidal structure whose unit is the delta sheaf at the origin or zero matrix. Variants of exactly this form appear in constructible sheaf theory, in microlocal and persistence settings, and in the additive Springer-theoretic setting on $\mathfrak{gl}_n$ (Benchoufi, 26 Apr 2026, Kashiwara et al., 2017, Bezrukavnikov et al., 2022).

A second model replaces addition by multiplication in a commutative group or group variety. On an abelian variety $A$, convolution is

$K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$

where $m:A\times A\to A$ is the group law. On $G=\mathbb{C}^*$, multiplicative convolution of perverse sheaves is

$\mathfrak{gl}_n$0

In both cases the group structure on the base supplies the monoidal product, but the categorical consequences differ: Tannakian formalism on abelian varieties and vanishing-cycle tensor structures on $\mathfrak{gl}_n$1 (Liu, 23 Jan 2025, Markarian, 29 Apr 2026).

A third model uses groupoid composition. For an ample groupoid $\mathfrak{gl}_n$2 with a $\mathfrak{gl}_n$3-sheaf of rings $\mathfrak{gl}_n$4, convolution is defined on compactly supported sheaf-valued sections by

$\mathfrak{gl}_n$5

Here the sheaf action transports coefficients between stalks before multiplication, and the result is a convolution algebra with nonconstant local coefficients (Gonçalves et al., 2019).

A fourth model appears in sheaf neural networks. There, “sheaf convolution” generally means propagation by a sheaf Laplacian $\mathfrak{gl}_n$6, so that the convolution layer is implemented by a diffusion step such as

$\mathfrak{gl}_n$7

In this usage the term refers not to pushforward along a geometric group law but to Laplacian-based message passing on graph-indexed stalks (Peng et al., 22 Apr 2026).

2. Geometric and representation-theoretic constructions

In the additive setting of the general linear Lie algebra, convolution is defined on $\mathfrak{gl}_n$8 by the addition map, yielding a symmetric monoidal structure whose unit is the delta sheaf at $\mathfrak{gl}_n$9. For the maximal parabolic $s$0 of type $s$1, the associated parabolic Springer sheaf is identified with $s$2, where $s$3 is the permutation representation of $s$4, and the paper proves

$s$5

A Fourier–Deligne transform intertwines additive convolution with a shifted tensor product, and exchanges $s$6 with the corresponding Grothendieck–Springer sheaf $s$7. This realizes representation-theoretic exterior powers as exterior powers inside a convolution monoidal category (Bezrukavnikov et al., 2022).

On abelian varieties, convolution plays a different role. Because $s$8 interacts well enough with the perverse $s$9-structure after quotienting by the Serre subcategory of Euler-characteristic-zero perverse sheaves, the quotient category $a$0 becomes Tannakian. For a relative perverse sheaf $a$1 on a constant abelian scheme $a$2, Liu studies the monodromy of

$a$3

and proves that for uncountably many character sheaves $a$4, the resulting monodromy group is a closed reductive normal subgroup of the generic convolution group $a$5. In this setting, convolution is the tensor structure underlying the generic symmetry group (Liu, 23 Jan 2025).

A multiplicative analogue is developed for perverse sheaves on $a$6. With

$a$7

vanishing cycles at $a$8 define a fiber functor compatible with convolution: $a$9 Markarian uses this compatibility, together with semi-holonomy isomorphisms and the geometry of $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$0 and $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$1, to relate multiplicative convolution to the harmonic coproduct and to prove that a homological pentagon condition is equivalent to the regularized double shuffle relations (Markarian, 29 Apr 2026).

A further variation is the “convolution direct image” for a finite Galois étale map $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$2 of commutative group schemes. Starting from the exact convolution bifunctor

$$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$3

on $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$4, Rojas-León defines a descent construction parallel to tensor direct image, now with convolution replacing tensor product. This yields an object on $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$5 obtained from the convolution of the $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$6 over the Galois group, and the resulting functor is used in arithmetic applications to exponential sums and point counting (Rojas-León, 2018).

3. Convolution algebras, kernels, and localization

In algebraic and noncommutative settings, sheaf convolution often means an algebra of sections with locally varying coefficients. For an ample groupoid $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$7 and a $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$8-sheaf of rings $$F\star G = Rs_!(F\boxtimes G) \quad\text{or}\quad F\star G = a_!(\pi_1^*F\otimes \pi_2^*G).$$9, the convolution algebra $\mathfrak{gl}_n$0 consists of finite sums of sections supported on compact open bisections, with product

$\mathfrak{gl}_n$1

This extends Steinberg algebras from constant coefficients to sheaf coefficients and is shown to be equivalent to skew inverse semigroup rings arising from spectral actions; conversely, every such sheaf convolution algebra can be presented as a skew inverse semigroup ring (Gonçalves et al., 2019).

For proper Lie groupoids, convolution is organized sheaf-theoretically over the orbit space. The convolution sheaf $\mathfrak{gl}_n$2 assigns to an open $\mathfrak{gl}_n$3 the algebra of smooth functions on the corresponding part of the arrow space whose support is longitudinally compact, with multiplication given by the usual groupoid convolution. The associated Hochschild chain presheaves are sheafified to obtain a Hochschild homology sheaf $\mathfrak{gl}_n$4, and the global Hochschild homology of the convolution algebra is identified with the global sections of that sheaf. A localization theorem then shows that the stalk at an orbit is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the linearized model $\mathfrak{gl}_n$5, reducing the local structure to compact group actions on vector spaces (Pflaum et al., 2020).

In microlocal sheaf theory, convolution is frequently an integral transform by a kernel. For $\mathfrak{gl}_n$6, the associated functor

$\mathfrak{gl}_n$7

gives a cocontinuous functor $\mathfrak{gl}_n$8, and Kuo–Li’s equivalence identifies such kernels with all cocontinuous functors between the microlocal categories. Hu proves that this convolution preserves compact objects if and only if every fiber $\mathfrak{gl}_n$9 is compact in $A$0, thereby classifying compact-preserving cocontinuous functors in kernel terms (Hu, 4 Nov 2025).

On a real vector space $A$1, constructible sheaves form another convolution algebra under

$A$2

Benchoufi studies invertible objects in this monoidal category and proves that the inverse of an invertible constructible sheaf $A$3 is $A$4, the Verdier dual of its antipodal transform. For compactly supported constant sheaves, invertibility is equivalent to convexity of the support. The paper also defines a microlocal transform $A$5, obtained by projecting the characteristic cycle of $A$6 to $A$7, and proves the compatibility relation

$A$8

which yields a necessary condition for invertibility (Benchoufi, 26 Apr 2026).

4. Persistence, distances, and higher-dimensional barcodes

In persistent homology and its sheaf-theoretic reformulations, convolution supplies both algebraic operations and metrics. Kashiwara–Schapira’s convolution on sheaves over a real vector space is

$A$9

and special kernels $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$0 obtained from balls define a pseudo-distance by $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$1-interleavings in the derived category. The resulting convolution distance satisfies a stability theorem for direct images: $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$2 When a closed convex proper cone $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$3 is fixed, the same framework is used to study $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$4-sheaves, their approximation by piecewise linear $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$5-sheaves, and higher-dimensional barcode-like stratifications (Kashiwara et al., 2017).

A related but distinct development treats persistence modules directly as sheaves and cosheaves on $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$6 with Alexandrov topologies. Two convolution operations are defined: $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$7 The first is canonically isomorphic to the derived tensor product of graded persistence modules, and the second admits a right-adjoint transform $K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$8. For one-parameter interval modules, the paper gives explicit formulas for the resulting degree-$K * L := Rm_*(p_1^*K \otimes^L p_2^*L),$9 and degree-$m:A\times A\to A$0 interval summands. Using the translation kernels $m:A\times A\to A$1 and $m:A\times A\to A$2, a convolution distance is defined and shown to coincide with the classical interleaving distance on persistence modules concentrated in degree $m:A\times A\to A$3 (Milicevic, 2020).

The sheaf-function correspondence produces a different kind of rigidity. Via the isomorphism between the Grothendieck group $m:A\times A\to A$4 and the group $m:A\times A\to A$5 of constructible functions, Asashiba, Kashiwara, Schapira, and Vilonen ask which distances on constructible functions are controlled by the sheaf convolution distance. Their main result is that any $m:A\times A\to A$6-dominated pseudo-extended metric on compactly supported constructible functions vanishes whenever the Euler integrals agree. In this precise sense, the induced distances are “almost trivial,” and a TDA consequence is that there cannot exist non-trivial additive invariants of persistence modules that are continuous for the interleaving distance (Berkouk, 2022).

5. Sheaf convolution in geometric deep learning

In graph-based machine learning, “sheaf convolution” usually denotes Laplacian propagation on a cellular sheaf. For a graph sheaf $m:A\times A\to A$7 with coboundary $m:A\times A\to A$8, the sheaf Laplacian is $m:A\times A\to A$9, and a standard Euclidean sheaf neural network layer is

$G=\mathbb{C}^*$0

The restriction maps are edge-specific linear maps, so the resulting message passing is anisotropic and heterogeneity-aware rather than adjacency-homogeneous (Peng et al., 22 Apr 2026).

This framework has been extended in several directions. In Bayesian Sheaf Neural Networks, the sheaf itself is learned variationally, with restriction maps modeled by distributions in $G=\mathbb{C}^*$1, by diagonal models, or by a Cayley-transform-based family of reparameterizable distributions on $G=\mathbb{C}^*$2. The resulting convolution is still the sheaf-diffusion layer based on the normalized sheaf Laplacian, but now with a distribution over sheaves rather than a single learned sheaf. The paper reports that Bayesian sheaf models outperform deterministic sheaf models when training data is limited and are less sensitive to hyperparameters (Gillespie et al., 2024).

A topological variant constructs the sheaf from local homology rather than from learned linear maps. Given the flag complex of a graph or a weighted Vietoris–Rips filtration, the local homology sheaf and its persistent version assign to each neighborhood a local homology vector space, with restriction maps derived from Mayer–Vietoris. The associated sheaf Laplacian then defines a convolution on persistent local homology features. This addresses two obstacles simultaneously: the lack of a natural basis for local homology groups and the need for differentiability with respect to weights in the filtration (Cesa et al., 2023).

The most geometric recent extension replaces Euclidean stalks by the manifold $G=\mathbb{C}^*$3. The core observation is that $G=\mathbb{C}^*$4 admits an abelian Lie group structure via

$G=\mathbb{C}^*$5

which allows intrinsic analogues of coboundary, adjoint, Laplacian, and convolution. Restriction maps are orthogonal congruences $G=\mathbb{C}^*$6, the coboundary is defined in the Lie algebra by log–exp transport, and the sheaf convolution update is

$G=\mathbb{C}^*$7

The paper proves that SPD-valued sheaves are strictly more expressive than Euclidean sheaves and reports state of the art on $G=\mathbb{C}^*$8 MoleculeNet benchmarks (Peng et al., 22 Apr 2026).

At the same time, the empirical necessity of learning nontrivial sheaf Laplacians has been challenged. The Identity Sheaf Network baseline fixes all restriction maps to the identity and shows comparable performance to several learned-sheaf architectures on five heterophilic benchmarks. Using Rayleigh quotients as a normalized measure of oversmoothing, the authors argue that the diffusion-based justification for learnable sheaf Laplacians is not reflected empirically in trained models. This establishes a concrete controversy inside the neural literature: the expressive theory of sheaf convolution does not automatically imply that learned restriction maps are the source of observed gains (Caralt et al., 5 Mar 2026).

6. Structural themes, limitations, and recurrent misconceptions

Across these settings, several structural motifs recur. One is the passage from a local external tensor product to a global composition law: addition on vector spaces and Lie algebras, multiplication on groups, composition in groupoids, or restriction-and-aggregation on graphs. Another is the presence of a distinguished unit, such as the delta sheaf at $G=\mathbb{C}^*$9, the skyscraper at the identity, the constant sheaf on the zero object, or the identity translation kernel. A third is the appearance of adjoint or dual operations—Fourier–Deligne transform for additive Springer theory, internal-Hom transforms in persistence, Verdier duality plus antipodal transform for invertible constructible sheaves, and compactness-preserving kernel criteria in microlocal categories (Bezrukavnikov et al., 2022, Milicevic, 2020, Benchoufi, 26 Apr 2026, Hu, 4 Nov 2025).

A common misconception is that “sheaf convolution” names a single standard construction. The surveyed literature instead uses the term for non-equivalent operations sharing a family resemblance: pushforward along a group law, composition in a groupoid with coefficient transport, kernel integral transforms, or Laplacian diffusion. This suggests that the phrase is best understood as a schema rather than a universally fixed definition.

The term also carries different kinds of mathematical output. In representation theory it can encode tensor categories and exterior power operations; in noncommutative geometry it yields convolution algebras and Hochschild homology sheaves; in persistence it generates interleaving-type metrics and approximation theorems; in neural architectures it supplies anisotropic message passing and topologically structured feature transport. Conversely, some constructions have sharp limitations. Distances on constructible functions controlled by sheaf convolution collapse to the Euler integral (Berkouk, 2022), and in neural applications the benefit of learnable restriction maps is not settled (Caralt et al., 5 Mar 2026).

A plausible implication is that future work will continue to separate genuinely geometric uses of sheaf convolution from merely formal analogies. The existing literature already points in that direction through precise criteria for compact-preserving kernels, microlocal obstructions to invertibility, local-to-global linearization theorems, and manifold-valued generalizations of sheaf diffusion (Hu, 4 Nov 2025, Benchoufi, 26 Apr 2026, Pflaum et al., 2020, Peng et al., 22 Apr 2026).