Cohomology for small categories: K-graphs and groupoids

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