ISOMORPHISM OF THE CUBICAL AND CATEGORICAL COHOMOLOGY GROUPS OF A HIGHER-RANK GRAPH

We use category-theoretic techniques to provide two proofs show-ing that for a higher-rank graph Λ, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we con-struct, explicitly and in both directions, maps on the level of (co-)chain com-plexes that implement said isomorphism. Along the way, we extend the defi-nition of cubical (co-)homology to allow arbitrary coefficient modules.