{"paper":{"title":"Cohomology for small categories: $k$-graphs and groupoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.OA","authors_text":"Alexander Kumjian, Elizabeth Gillaspy","submitted_at":"2015-11-03T20:35:50Z","abstract_excerpt":"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_\\L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.01073","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}