Results of Fowler and Sims show that every k-graph is completely determined
by its k-coloured skeleton and collection of commuting squares. Here we give an
explicit description of the k-graph associated to a given skeleton and
collection of squares and show that two k-graphs are isomorphic if and only if
there is an isomorphism of their skeletons which preserves commuting squares.
We use this to prove directly that each k-graph {\Lambda} is isomorphic to the
quotient of the path category of its skeleton by the equivalence relation
determined by the commuting squares, and show that this extends to a
homeomorphism of infinite-path spaces when the k-graph is row finite with no
sources. We conclude with a short direct proof of the characterisation,
originally due to Robertson and Sims, of simplicity of the C*-algebra of a
row-finite k-graph with no sources.Comment: 21 pages, two pictures prepared using TiK
