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A survey of the Preservation of Symmetries by the Dual Gromov-Hausdorff Propinquity

We survey the symmetry preserving properties for the dual propinquity, under natural non-degeneracy and equicontinuity conditions. These properties are best formulated using the notion of the covariant propinquity when the symmetries are encoded via the actions of proper monoids and groups. We explore the issue of convergence of Cauchy sequences for the covariant propinquity, which captures, via a compactness result, the fact that proper monoid actions can pass to the limit for the dual propinquity.