Open Condensed Subgroups and Mackey's Formula

TL;DR

This paper extends classical results about modules to the setting of condensed groups, defining open actions and establishing Mackey's formula, with potential for further solidification of these results.

Contribution

It introduces the concept of open condensed group actions and proves Mackey's formula in this new context, bridging abstract module theory and condensed mathematics.

Findings

Open condensed group actions are well-defined following Scholze.

Mackey's formula holds for modules over open condensed subgroups.

Results can be solidified to obtain solid versions.

Abstract

We define what it means for a condensed group action to be open (following Scholze) and show that for open subgroups, many elementary results about abstract modules hold for condensed modules, such as the existence of Mackey's Formula for condensed groups. We also indicate how these results can be "solidified" to obtain their solid versions.