Partial Category Actions on Sets and Topological Spaces

TL;DR

This paper develops the theory of partial category actions on sets and topological spaces, establishing universal globalizations and generalizing previous results for groups and monoids, with applications to partial groupoid actions.

Contribution

It introduces a framework for partial category actions with universal globalizations, extending known results for groups and monoids to broader categorical contexts.

Findings

Every partial category action admits a universal globalization.

Generalization of group and monoid action results to category actions.

Injectivity of mediating functions in partial groupoid actions.

Abstract

We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Kellendonk and Lawson, and for monoids, by Megrelishvili and Schroder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.