Generalised morphisms of 𝑘-graphs: 𝑘-morphs

Kumjian, Alex; Pask, David; Sims, Aidan

doi:10.1090/S0002-9947-2010-05152-9

Generalised morphisms of $k$-graphs: $k$-morphs
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Trans. Amer. Math. Soc. 363 (2011), 2599-2626 Request permission

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 $\Lambda \mapsto C^*(\Lambda )$ to a functor from this category to the category whose objects are $C^*$-algebras and whose morphisms are isomorphism classes of $C^*$-correspondences.

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