Cohomology for small categories: k-graphs and groupoids

Abstract

Given a higher-rank graph $\Lambda$, we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid ${\mathcal{G}}_{\Lambda }$. We define an exact functor between the Abelian category of right modules over a higher-rank graph $\Lambda$ and the category of ${\mathcal{G}}_{\Lambda }$-sheaves, where ${\mathcal{G}}_{\Lambda }$ is the path groupoid of $\Lambda$. We use this functor to construct compatible homomorphisms from both the cohomology of $\Lambda$ with coefficients in a right $\Lambda$-module, and the continuous cocycle cohomology of ${\mathcal{G}}_{\Lambda }$ with values in the corresponding ${\mathcal{G}}_{\Lambda }$-sheaf, into the sheaf cohomology of ${\mathcal{G}}_{\Lambda }$.