Generalised morphisms of 𝑘-graphs: 𝑘-morphs

Abstract

Abstract. 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 Λ → C * (Λ) to a functor from this category to the category whose objects are C * -algebras and whose morphisms are isomorphism classes of C * -correspondences.

On twisted higher-rank graph 𝐶*-algebras

“…We adopt the conventions of [13,16] for k-graphs. Given a nonnegative integer k, a k-graph is a nonempty countable small category Λ equipped with a functor d : Λ → N k satisfying the factorisation property: for all λ ∈ Λ and m, n ∈ N k such that d(λ) = m + n there exist unique μ, ν ∈ Λ such that d(μ) = m, d(ν) = n, and λ = μν.…”

Groupoids and $C^*$-algebras for categories of paths

“…(It is not clear how to build a k-graph from a collection of k commuting graphs. See [18], Example 5.15(ii). The construction we give treats this example as easily as a finite k-graph.)…”

Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras

“…By contrast, [17,Example 5.15(ii)] presents an example, due to Jack Spielberg, of a 3-skeleton that cannot be the skeleton of a 3-graph, and it is straightforward to extend the arguments used in that example to see that this 3-skeleton does not admit any unitary cocycles. In particular, for k ≥ 3 the existence of a unitary cocycle for (E 1 , .…”