Monadic functors forgetful of (dis)inhibited actions

TL;DR

This paper establishes that for various categories of topological or uniform spaces with group actions, the forgetful functor to the base category is monadic, enabling unified results on equivariant completions and compactifications.

Contribution

It generalizes and unifies existing results on monadicity and cocompleteness of categories of group actions on topological and uniform spaces.

Findings

The forgetful functor is monadic in many topological and uniform space categories.

Categories of G-flows are cocomplete, allowing for equivariant completions.

Results recover and extend prior work on equivariant compactifications.

Abstract

We prove a number of results of the following common flavor: for a category $\mathcal{C}$ of topological or uniform spaces with all manner of other properties of common interest (separation / completeness / compactness axioms), a group (or monoid) $\mathbb{G}$ equipped with various types of topological structure (topologies, uniformities) and the corresponding category $\mathcal{C}^{\mathbb{G}}$ of appropriately compatible $\mathbb{G}$-flows in $\mathcal{C}$, the forgetful functor $\mathcal{C}^{\mathbb{G}}\to \mathcal{C}$ is monadic. In all cases of interest the domain category $\mathcal{C}^{\mathbb{G}}$ is also cocomplete, so that results on adjunction lifts along monadic functors apply to provide equivariant completion and/or compactification functors. This recovers, unifies and generalizes a number of such results in the literature due to de Vries, Mart'yanov and others on existence of equivariant compactifications / completions and cocompleteness of flow categories.