Generalised morphisms of 𝑘-graphs: 𝑘-morphs

In a number of recent papers, ( k + l ) (k+l) -graphs have been constructed from k k -graphs by inserting new edges in the last l l dimensions. These constructions have been motivated by C ∗ C^* -algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k k -morphs, which provide a systematic unifying framework for these various constructions. We think of k k -morphs as the analogue, at the level of k k -graphs, of C ∗ C^* -correspondences between C ∗ C^* -algebras. To make this analogy explicit, we introduce a category whose objects are k k -graphs and whose morphisms are isomorphism classes of k k -morphs. We show how to extend the assignment Λ ↦ C ∗ ( Λ ) Λ ↦ C^*(Λ ) to a functor from this category to the category whose objects are C ∗ C^* -algebras and whose morphisms are isomorphism classes of C ∗ C^* -correspondences.