{"paper":{"title":"Decomposition Spaces, Incidence Algebras and M\\\"obius Inversion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.CT","authors_text":"Andrew Tonks, Imma G\\'alvez-Carrillo, Joachim Kock","submitted_at":"2014-04-11T19:46:45Z","abstract_excerpt":"We introduce the notion of decomposition space as a general framework for incidence algebras and M\\\"obius inversion: it is a simplicial infinity-groupoid satisfying an exactness condition weaker than the Segal condition, which expresses decomposition. We work on the objective level of homotopy linear algebra with coefficients in infinity-groupoids, developed along the way. To any (complete) decomposition space there is associated an incidence (co)algebra (with coefficients in infinity-groupoids), shown to satisfy a sign-free version of the M\\\"obius inversion principle. Examples of decompositio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3202","kind":"arxiv","version":5},"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"}