  Topological data analysis is a powerful tool for describing topological
signatures in real world data. An important challenge in topological data
analysis is matching significant topological signals across distinct systems.
In geometry and probability theory, optimal transport formalises notions of
distance and matchings between distributions and structured objects. We propose
to combine these approaches, constructing a mathematical framework for optimal
transport-based matchings of topological features. Building upon recent
advances in the domains of persistent homology and optimal transport for
hypergraphs, we develop a transport-based methodology for topological data
processing. We define measure topological networks, which integrate both
geometric and topological information about a system, introduce a distance on
the space of these objects, and study its metric properties, showing that it
induces a geodesic metric space of non-negative curvature. The resulting
Topological Optimal Transport (TpOT) framework provides a transport model on
point clouds that minimises topological distortion while simultaneously
yielding a geometrically informed matching between persistent homology cycles.
