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 $G_\Lambda$. We define an exact functor between the abelian category of right modules over a higher-rank graph $\Lambda$ and the category of $G_\Lambda$-sheaves, where $G_\Lambda$ 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 $G_\Lambda$ with values in the corresponding $G_\Lambda$-sheaf, into the sheaf cohomology of $G_\Lambda$.