  Higher-order relations are widespread in nature, with numerous phenomena
involving complex interactions that extend beyond simple pairwise connections.
As a result, advancements in higher-order processing can accelerate the growth
of various fields requiring structured data. Current approaches typically
represent these interactions using hypergraphs. We enhance this representation
by introducing cellular sheaves for hypergraphs, a mathematical construction
that adds extra structure to the conventional hypergraph while maintaining
their local, higherorder connectivity. Drawing inspiration from existing
Laplacians in the literature, we develop two unique formulations of sheaf
hypergraph Laplacians: linear and non-linear. Our theoretical analysis
demonstrates that incorporating sheaves into the hypergraph Laplacian provides
a more expressive inductive bias than standard hypergraph diffusion, creating a
powerful instrument for effectively modelling complex data structures. We
employ these sheaf hypergraph Laplacians to design two categories of models:
Sheaf Hypergraph Neural Networks and Sheaf Hypergraph Convolutional Networks.
These models generalize classical Hypergraph Networks often found in the
literature. Through extensive experimentation, we show that this generalization
significantly improves performance, achieving top results on multiple benchmark
datasets for hypergraph node classification.
