Cohomology for small categories: $k$ -graphs and groupoids
| Autor: | Elizabeth Gillaspy, Alex Kumjian |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Path (topology)
Sheaf cohomology Pure mathematics 18G60 (Primary) 22A22 55N30 16E30 18B40 (Secondary) 46L05 Mathematics::Algebraic Topology 01 natural sciences 55N30 Mathematics::K-Theory and Homology Mathematics::Category Theory 0103 physical sciences FOS: Mathematics Category Theory (math.CT) 0101 mathematics Operator Algebras (math.OA) 22E41 Mathematics Algebra and Number Theory Functor groupoids higher-rank graphs 010102 general mathematics Mathematics - Operator Algebras Mathematics - Category Theory Graph Cohomology 18B40 cohomology Homomorphism 010307 mathematical physics Abelian category Exact functor Analysis |
| Zdroj: | Banach J. Math. Anal. 12, no. 3 (2018), 572-599 |
| ISSN: | 1735-8787 |
| DOI: | 10.1215/17358787-2017-0041 |
| Popis: | 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 $G_\Lambda$. Comment: A flaw in the proof of Proposition 4.2 in v1 of this paper has invalidated Proposition 4.8 and Theorem 4.9 from v1. This version (v3) has been substantially revised and includes new results. Version 4 to appear in Banach J. Math. Anal |
| Databáze: | OpenAIRE |
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