{"paper":{"title":"Topos Semantics for Higher-Order Modal Logic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.LO","authors_text":"Hans-Christoph Kotzsch, Kohei Kishida, Steve Awodey","submitted_at":"2014-02-28T22:05:11Z","abstract_excerpt":"We define the notion of a model of higher-order modal logic in an arbitrary elementary topos $\\mathcal{E}$. In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier $\\Omega_{\\mathcal{E}}$, but rather by a suitable complete Heyting algebra $H$. The canonical map relating $H$ and $\\Omega_{\\mathcal{E}}$ both serves to interpret equality and provides a modal operator on $H$ in the form of a comonad. Examples of such structures arise from surjective geometric morphisms $f : \\mathcal{F} \\to \\mathcal{E}$, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0020","kind":"arxiv","version":1},"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"}