  A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an
embedding of universal C*-algebras. We show how to build a (k+1)-graph whose
universal algebra encodes this embedding. More generally we show how to realise
a direct limit of k-graph algebras under embeddings induced from coverings as
the universal algebra of a (k+1)-graph. Our main focus is on computing the
K-theory of the (k+1)-graph algebra from that of the component k-graph
algebras.
  Examples of our construction include a realisation of the Kirchberg algebra
\mathcal{P}_n whose K-theory is opposite to that of \mathcal{O}_n, and a class
of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens
algebras.
