{"paper":{"title":"Grothendieck topologies on a poset","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CT","authors_text":"Bert Lindenhovius","submitted_at":"2014-05-17T15:06:55Z","abstract_excerpt":"We investigate Grothendieck topologies (in the sense of sheaf theory) on a poset $\\P$ that are generated by some subset of $\\P$. We show that such Grothendieck topologies exhaust all possibilities if and only if $\\P$ is Artinian. If $\\P$ is not Artinian, other families of Grothendieck topologies on $\\P$ exist that are not generated by some subset of $\\P$, but even those are related to the Grothendieck topologies generated by subsets. Furthermore, we investigate several notions of equivalences of Grothendieck topologies, and using a posetal version of the Comparison Lemma, a sheaf-theoretic res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4408","kind":"arxiv","version":2},"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"}