Cell Poset

A cell poset is the common skeleton beneath graphs, simplicial complexes, CW complexes, and hypergraphs. Forgetting the geometry and keeping only the cells and the relation “is a face of” yields a partially-ordered-set, a Hasse diagram whose -cells sit above the -cells they bound. One machine then describes message passing on all four kinds of structure at once, rather than four bespoke schemes.

Concretely, a cell poset is a poset whose elements are cells graded by dimension, with meaning is a face of . A graph gives vertices below edges; a simplicial complex adds triangles above their edges; a CW complex attaches cells of every dimension; a hypergraph puts each hyperedge above the vertices it contains. The face/incidence order is the same in each case.

A feature assignment is then a functor on the poset, sending each cell to its data and each incidence to a transport map. When that functor is -valued, carrying restriction maps between stalks, it is exactly a cellular-sheaf on .

Definition. Aggregation as a weighted colimit

Message passing over a cell poset is a weighted colimit computed on the functor of features. Forming the colimit over the category of elements of glues the per-cell data along incidences, and the weight records how strongly each face contributes. Graph convolution, simplicial and CW message passing, and hypergraph aggregation are all this one colimit read on different posets.

Example. Graph as a cell poset

A graph is the cell poset with vertices at grade and edges at grade , each edge covering its two endpoints. A feature functor places node and edge features as stalks, and the colimit over the incidence poset is ordinary neighbourhood aggregation. Enlarging the poset with -cells upgrades the same construction to simplicial message passing with no change of machinery.

References

• Hajij et al., “Topological Deep Learning: Going Beyond Graph Data” (2022)