  We examine overlapping clustering schemes with functorial constraints, in the
spirit of Carlsson--Memoli. This avoids issues arising from the chaining
required by partition-based methods. Our principal result shows that any
clustering functor is naturally constrained to refine single-linkage clusters
and be refined by maximal-linkage clusters. We work in the context of metric
spaces with non-expansive maps, which is appropriate for modeling data
processing which does not increase information content.
