Banach Journal of Mathematical Analysis
- Banach J. Math. Anal.
- Volume 12, Number 3 (2018), 572-599.
Cohomology for small categories: -graphs and groupoids
Given a higher-rank graph , we investigate the relationship between the cohomology of and the cohomology of the associated groupoid . We define an exact functor between the Abelian category of right modules over a higher-rank graph and the category of -sheaves, where is the path groupoid of . We use this functor to construct compatible homomorphisms from both the cohomology of with coefficients in a right -module, and the continuous cocycle cohomology of with values in the corresponding -sheaf, into the sheaf cohomology of .
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.
Banach J. Math. Anal., Volume 12, Number 3 (2018), 572-599.
Received: 9 February 2017
Accepted: 2 May 2017
First available in Project Euclid: 10 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) [See also 20Axx, 20L05, 20Mxx]
Secondary: 55N30: Sheaf cohomology [See also 18F20, 32C35, 32L10] 22E41: Continuous cohomology [See also 57R32, 57Txx, 58H10] 46L05: General theory of $C^*$-algebras
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