  Collections of measures on compact metric spaces form a model category ("data
complexes"), whose morphisms are marginalization integrals. The fibrant objects
in this category represent collections of measures in which there is a measure
on a product space that marginalizes to any measures on pairs of its factors.
The homotopy and homology for this category allow measurement of obstructions
to finding measures on larger and larger product spaces. The obstruction theory
is compatible with a fibrant filtration built from the Wasserstein distance on
measures.
  Despite the abstract tools, this is motivated by a widespread problem in data
science. Data complexes provide a mathematical foundation for semi-automated
data-alignment tools that are common in commercial database software.
Practically speaking, the theory shows that database JOIN operations are
subject to genuine topological obstructions. Those obstructions can be detected
by an obstruction cocycle and can be resolved by moving through a filtration.
Thus, any collection of databases has a persistence level, which measures the
difficulty of JOINing those databases. Because of its general formulation, this
persistent obstruction theory also encompasses multi-modal data fusion
problems, some forms of Bayesian inference, and probability couplings.
