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There has been an explosion of interest in a new type of topological phase of quantum matter, where the object of study is not a single many-body system but an entire family of them, depending continuously on parameters in a topological space. Starting from a parametrized one-dimensional spin system, I will show how C*-algebra theory can be used to derive a phase invariant valued in the third cohomology group of the parameter space. This construction will be exhibited for an exactly solvable model in a nontrivial phase parametrized by the 3-sphere.