The path space of a higher-rank graph

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\mathrm{\Lambda }$. We identify the boundary-path space $\mathrm{\partial }\mathrm{\Lambda }$ as the spectrum of a commutative $C^{\ast }$-subalgebra $D_{\mathrm{\Lambda }}$ of $C^{\ast }(\mathrm{\Lambda })$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph $\text{wt}\mathrm{\Lambda }$ with no sources in which $\mathrm{\Lambda }$ is embedded, and show that $\mathrm{\partial }\mathrm{\Lambda }$ is homeomorphic to a subset of $\mathrm{\partial }\text{wt}\mathrm{\Lambda }$ . We show that when $\mathrm{\Lambda }$ is row-finite, we can identify $C^{\ast }(\mathrm{\Lambda })$ with a full corner of $C^{\ast }(\text{wt}\mathrm{\Lambda })$, and deduce that $D_{\mathrm{\Lambda }}$ is isomorphic to a corner of $D_{\text{wt}\mathrm{\Lambda }}$. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.