Partial Category Actions on Sets and Topological Spaces

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.