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 Pn whose K-theory is opposite to that of On, and a class of AT-algebras that can naturally be regarded as higher-rank Bunce- Deddens algebras.