  We bring together topological data analysis, applied category theory, and
machine learning to study multiparameter hierarchical clustering. We begin by
introducing a procedure for flattening multiparameter hierarchical clusterings.
We demonstrate that this procedure is a functor from a category of
multiparameter hierarchical partitions to a category of binary integer
programs. We also include empirical results demonstrating its effectiveness.
Next, we introduce a Bayesian update algorithm for learning clustering
parameters from data. We demonstrate that the composition of this algorithm
with our flattening procedure satisfies a consistency property.
