Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph
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- by Elizabeth Gillaspy and Jianchao Wu;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 442-480
- DOI: https://doi.org/10.1090/btran/38
- Published electronically: June 10, 2021
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Abstract:
We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $\Lambda$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.References
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Bibliographic Information
- Elizabeth Gillaspy
- Affiliation: Department of Mathematical Sciences, University of Montana, 32 Campus Drive #0864, Missoula, Montana 59812
- MR Author ID: 1107754
- Email: elizabeth.gillaspy@mso.umt.edu
- Jianchao Wu
- Affiliation: Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, Texas 77843
- MR Author ID: 1182142
- ORCID: 0000-0002-4937-763X
- Email: jwu@tamu.edu
- Received by editor(s): July 19, 2018
- Received by editor(s) in revised form: January 26, 2019
- Published electronically: June 10, 2021
- Additional Notes: The first author was partially supported by the Deutsches Forschungsgemeinschaft via the SFB 878 “Groups, Geometry, and Actions.” The second author was partially supported by NSF grant #DMS–1564401.
- © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 442-480
- MSC (2020): Primary 18G90; Secondary 55N10
- DOI: https://doi.org/10.1090/btran/38
- MathSciNet review: 4273194