Generalised morphisms of k-graphs: k-morphs

In a number of recent papers, $(k+l)$-graphs have been

constructed from $k$-graphs by inserting new edges in the last

$l$ dimensions. These constructions have been motivated by

$C^*$-algebraic considerations, so they have not been treated

systematically at the level of higher-rank graphs themselves.

Here we introduce $k$-morphs, which provide a systematic

unifying framework for these various constructions. We think of

$k$-morphs as the analogue, at the level of $k$-graphs, of

$C^*$-correspondences between $C^*$-algebras. To make this

analogy explicit, we introduce a category whose objects are

$k$-graphs and whose morphisms are isomorphism classes of

$k$-morphs. We show how to extend the assignment $\Lambda \mapsto

C^*(\Lambda)$ to a functor from this category to the category

whose objects are $C^*$-algebras and whose morphisms are

isomorphism classes of $C^*$-correspondences.