Sheaf Laplacians and the Shape of the Base

Take the sheaf-neural-network base to be the category of elements of a graph-presheaf; then building its message-passing sheaf Laplacian forces conditions on the shape of that base. Which shapes admit a Laplacian at all, and does the shape select the calculus?

topological-deep-learning fixes the established setup. A directed graph is a presheaf on the walking-graph category, its category of elements is the base, and a feature sheaf separates the structural presheaf from the feature sheaf. The questions below ask what that base must be for the sheaf Laplacian to exist and what its shape buys. Orthogonally, the layers compose as an operad of NN blocks.

When does a sheaf Laplacian exist at all?

The cochains must form a complex, a graded direct sum of stalks equipped with a coboundary satisfying . That nullity is the constraint. It requires the base to be combinatorially nice enough to carry a orientation and a grading, as met by the simplex category or anything Dold–Kan-equivalent to a chain complex. For a general element-poset, what plays the role of “-dimension”, and when is it even well defined through ranks or dependent pairs?

How far can nullity be relaxed?

Is there a shape category between and an arbitrary that still gives , such as a Reedy category whose dimension is an ordinal with raising and lowering maps? And when is a site, a Grothendieck coverage, so that is a sheaf rather than merely a presheaf of features? Homology, after all, may not need a topology at all.

Which shape selects which calculus?

Globular, simplicial, cubical, dendroidal, opetopic bases each yield a different Hodge/Laplacian, and the equivalences between them are largely open. This is the discrete face of the bet that the shape of a higher category selects the computation. The base shape fixes the differential calculus, hence the network’s inductive bias. The fragment-2-rdm-sheaf with stalks is one instance; a connection version adds curvature per edge.

Recent work bears on the first question. Sheaf cohomology now has an algorithm on any finite poset, and most notions previously tied to cell complexes transfer to arbitrary posets, so the base need not be a cell complex for the calculus to exist (Ayzenberg et al.). Maruyama’s order-equivariant networks fix one such shape, a face poset carrying a group action, and read off its equivariant calculus. A systematic benchmark of the sheaf design space finds the restriction map, rather than the base, dominates measured performance (Fiorini et al.), so whether the shape genuinely selects the calculus is still open empirically.

References

• Ayzenberg et al., sheaf cohomology on arbitrary finite posets (arXiv:2502.15476).
• Maruyama, order-equivariant neural networks (arXiv:2607.03798).
• F. Fiorini et al., a systematic benchmark of the sheaf design space (arXiv:2608.02558).