2014/05/17 by Bert Lindenhovius, Lindenhovius, Bert · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Category Theory (math.CT) #Commutative Algebra and Its Applications #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1405.4408
openalex publication_date 2014/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate Grothendieck topologies (in the sense of sheaf theory) on a poset ¶ that are generated by some subset of ¶. We show that such Grothendieck topologies exhaust all possibilities if and only if ¶ is Artinian. If ¶ is not Artinian, other families of Grothendieck topologies on ¶ exist that are not generated by some subset of ¶, 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 result known as the Comparison Lemma, going back to Grothendieck et al \citeSGA4, we calculate the sheaves with respect to most of the Grothendieck topologies we have found.