{"paper":{"title":"On two-Dimensional Holonomy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.CT"],"primary_cat":"math.DG","authors_text":"Joao Faria Martins, Roger Picken","submitted_at":"2007-10-23T18:39:51Z","abstract_excerpt":"We define the thin fundamental categorical group ${\\mathcal P}_2(M,*)$ of a based smooth manifold $(M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$, and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps are rank-$n$ homotopic, when the rank of the differential of the homotopy between them equals $n$. Let $\\C(\\Gc)$ be a Lie categorical group coming from a Lie crossed module ${\\Gc= (\\d\\colon E \\to G,\\tr)}$. We construct categorical holonomies, defined to be smooth morphisms ${\\mathcal P}_2(M,*) \\to \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.4310","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"}