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arXiv · 1511.01073

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

Elizabeth Gillaspy·Alexander Kumjian
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Abstract

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 $G_Λ$.

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Elizabeth Gillaspy, Alexander Kumjian. 2017-05-03. Cohomology for small categories: $k$-graphs and groupoids. https://doi.org/10.1215/17358787-2017-0041

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