Higher-rank graphs are certain small categories that are a higher-dimensional generalisation of both directed graphs and free monoids. Path and boundary-path groupoids of finitely aligned higher-rank graphs are often constructed using either filters or graph morphisms. We generalise the graph morphism approach to finitely aligned P-graphs where (Q, P) is a weakly quasi-lattice ordered group, and we show the filter approach and the graph morphism approach yield isomorphic path and boundary-path groupoids. To do this, we define conjugacy of partial monoid actions such that conjugate actions have isomorphic semidirect product groupoids. Combining our results with others in the literature, we survey many isomorphic presentations of path and boundary-path groupoids at different levels of generality.

This research was supported by Marsden grant 21-VUW-156 from the Royal Society of New Zealand and partially supported by a grant from the Simons Foundation. This work was also supported by EPSRC Grant Number EP/V521929/1, and the second author was supported by grant P1-0288 from the Slovenian Research and Innovation Agency. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for the support and hospitality during the programme "Topological groupoids and their C*-algebras" when work on this paper was undertaken.