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

TL;DR

This paper surveys how the dual Gromov-Hausdorff propinquity preserves symmetries, focusing on actions of monoids and groups, and discusses convergence properties of these symmetries in the noncommutative metric setting.

Contribution

It introduces the covariant propinquity to better understand symmetry preservation and convergence of symmetries under the dual propinquity in noncommutative geometry.

Findings

Symmetry preservation properties are characterized under non-degeneracy and equicontinuity.

Proper monoid actions can pass to the limit in the dual propinquity setting.

The covariant propinquity effectively captures the convergence of symmetries.

Abstract

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.