Actions of pro-groups and pro-rings

TL;DR

This paper explicitly describes internal actions in semi-abelian categories of pro-groups and pro-rings using actions in the pro-sets category, highlighting differences with Lie algebras.

Contribution

It provides a detailed description of internal actions in pro-group and pro-ring categories, extending the understanding of their structure.

Findings

Explicit description of internal actions in pro-groups and pro-rings

Demonstration of the failure of similar results for Lie algebras

Connection to actions of group and ring objects in pro-sets

Abstract

We give an explicit description of internal actions in the semi-abelian categories of pro-groups and non-unital pro-rings in terms of actions of group objects and ring objects in $\mathrm{Pro}(\mathbf{Set})$, as well as in some related categories. Also, we show that a similar result fails for Lie algebras.