Topological Deep Learning

Topological deep learning carries the computation on a complex rather than a graph, and its sharpest hard constraint is functorial. A cellular sheaf fixes restriction maps, so the network can only produce data that glues consistently across cells.

Topological deep learning runs a neural network over a structure richer than a set of pairwise edges, namely a simplicial or cell complex, or a sheaf on one. Higher-order incidence is therefore part of the substrate rather than something the model must rediscover. Its most structural instance is the sheaf neural network (Hansen–Gebhart; Bodnar et al.), where the constraint reads categorically. A cellular sheaf is a functor on the cell poset of the complex, assigning a stalk to each cell and a restriction map along each incidence. Fixing those restriction maps fixes a global consistency law, since the sheaf’s sections are exactly the assignments that agree across every incidence, and the network is confined to respect it. That is the hard constraint in its purest form, not a symmetry to commute with, but a functoriality to obey.

A cleaner base makes the split explicit. Present the graph as a presheaf on the walking-graph category ; its category of elements has vertices and edges as objects and incidences as morphisms, and a feature sheaf is a functor into feature spaces (, ). This separates the structure carried by the presheaf from the feature carried by the sheaf. By the coend calculus a feature sheaf is equivalently a graph homomorphism from into a category of local feature data. Message passing is then the sheaf Laplacian , the Hodge Laplacian of the sheaf cochain complex once the stalks carry inner products. Its Dirichlet energy measures cross-cell disagreement, so the inductive bias is, verbatim, minimise inconsistency with the fixed gluing.

Functoriality is the constraint

A graph neural network gets equivariance from naturality over incidence; a sheaf network tightens this to the sheaf’s own restriction maps, so the admissible computations are exactly the functorial ones over the sheaf’s base. Maruyama’s Foundations of Equivariant Deep Learning: Unifying Graph and Sheaf Neural Networks (2026) places graph- and sheaf-networks in one equivariant frame on exactly this reading, the unification pursued here; the open problem of which base shape (simplicial, cubical, Reedy, …) admits a Laplacian at all, and so selects the calculus, is pursued separately.