  Persistent homology is a powerful tool for characterizing the topology of a
data set at various geometric scales. When applied to the description of
molecular structures, persistent homology can capture the multiscale geometric
features and reveal certain interaction patterns in terms of topological
invariants. However, in addition to the geometric information, there is a wide
variety of non-geometric information of molecular structures, such as element
types, atomic partial charges, atomic pairwise interactions, and electrostatic
potential function, that is not described by persistent homology. Although
element specific homology and electrostatic persistent homology can encode some
non-geometric information into geometry based topological invariants, it is
desirable to have a mathematical framework to systematically embed both
geometric and non-geometric information, i.e., multicomponent heterogeneous
information, into unified topological descriptions. To this end, we propose a
mathematical framework based on persistent cohomology. In our framework,
non-geometric information can be either distributed globally or resided locally
on the datasets in the geometric sense and can be properly defined on
topological spaces, i.e., simplicial complexes. Using the proposed persistent
cohomology based framework, enriched barcodes are extracted from datasets to
represent heterogeneous information. We consider a variety of datasets to
validate the present formulation and illustrate the usefulness of the proposed
persistent cohomology. It is found that the proposed framework using cohomology
boosts the performance of persistent homology based methods in the
protein-ligand binding affinity prediction on massive biomolecular datasets.
