Cohomology for small categories: $k$-graphs and groupoids

TL;DR

This paper explores the relationship between the cohomology of higher-rank graphs and their associated groupoids, introducing a functor that links modules over graphs to sheaves over groupoids, and constructing compatible cohomology homomorphisms.

Contribution

It defines an exact functor connecting modules over higher-rank graphs to groupoid sheaves and constructs compatible cohomology homomorphisms, advancing the understanding of their cohomological relationship.

Findings

Established an exact functor between module categories and sheaf categories.

Constructed compatible homomorphisms from graph and groupoid cohomology to sheaf cohomology.

Provided a framework linking graph cohomology with groupoid cohomology.

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 $G_\Lambda$.