Monadic functors forgetful of (dis)inhibited actions
Alexandru Chirvasitu
TL;DR
The paper addresses unifying equivariant topological structures under group actions by showing the forgetful functor $\mathcal C^{\mathbb G} \to \mathcal C$ is monadic in broad settings, enabling adjunction lifts to equivariant completions. Treating a $\mathbb G$-flow as a $T$-algebra for a monad $T$ on $\mathcal C$, it applies Beck's Precise Tripleability and Freyd's Adjoint Functor Theorem to obtain monadicity and cocompleteness, which in turn yield left adjoints realizing equivariant compactifications and completions. Key results include wide monadicity and cocompleteness for $\mathcal C^{\mathbb M}_{\mathcal F}$ across $\mathcal C$ (e.g., Top, Unif, Cpct) and various action-constraints, plus lifting of reflective inclusions to flows. This framework unifies and extends known equivariant compactification/completion results by de Vries, Mart'yanov, and others, providing a robust categorical toolkit for constructing and universalizing equivariant completions.
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.