We prove a number of results of the following common flavor: for a category C of topological or uniform spaces with all manner of other properties of common interest (separation / completeness / compactness axioms), a group (or monoid) G equipped with various types of topological structure (topologies, uniformities) and the corresponding category C^G of appropriately compatible G-flows in C, the forgetful functor C^G → C is monadic. In all cases of interest the domain category C^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.
Keywords: Tychonoff space; adjoint functor theorem; bounded flow; closed category; cocomplete; compactification; compactly generated; completion; concrete category; enriched category; exponentiable; flow; internal group; jointly continuous; monadic; monoidal category; monoidal functor; quasi-bounded flow; reflective; separately continuous; solution-set condition; split coequalizer; tripleability; uniformity
2020 MSC: 18C15; 18D20; 18A30; 18A40; 54D35; 54E15; 54A20; 22F05
Theory and Applications of Categories, Vol. 41, 2024, No. 44, pp 1536-1556.
Published 2024-10-16.
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