## Banach Journal of Mathematical Analysis

### 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 $\mathcal{G}_{\Lambda}$.

#### Note

The current online version of this article, posted on 19 December 2017, supersedes the advance publication version posted on 10 November 2017. The affiliation and contact information for the first author have been corrected.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 3 (2018), 572-599.

Dates
Accepted: 2 May 2017
First available in Project Euclid: 10 November 2017

https://projecteuclid.org/euclid.bjma/1510283128

Digital Object Identifier
doi:10.1215/17358787-2017-0041

Mathematical Reviews number (MathSciNet)
MR3824741

Zentralblatt MATH identifier
06946071

#### Citation

Gillaspy, Elizabeth; Kumjian, Alexander. Cohomology for small categories: $k$ -graphs and groupoids. Banach J. Math. Anal. 12 (2018), no. 3, 572--599. doi:10.1215/17358787-2017-0041. https://projecteuclid.org/euclid.bjma/1510283128

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