Functorial Clustering via Simplicial Complexes

We adapt previous research on topological unsupervised learning to characterize hierarchical overlapping clustering algorithms as functors that factor through a category of simplicial complexes. We first develop a pair of adjoint functors that map between simplicial complexes and the outputs of clustering algorithms. Next, we introduce the maximal and single linkage clustering algorithms as the respective composition of the flagification and connected components functors with McInnes et al's finite singular set functor. We then adapt a theorem by Culbertson et al to demonstrate that all other hierarchical overlapping clustering functors are refined by maximal linkage and refine single linkage.