{"paper":{"title":"Fat Lie Theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Fat extensions of Lie groupoids correspond one-to-one with abstract 2-term representations up to homotopy.","cross_cats":["math.AT","math.RT","math.SG"],"primary_cat":"math.DG","authors_text":"Lennart Obster","submitted_at":"2026-03-09T09:57:03Z","abstract_excerpt":"We discuss a new point of view of representation theory of Lie groupoids and algebroids: fat Lie theory. The category of fat extensions is introduced, as well as the category of abstract $2$-term representations up to homotopy (ruths) -- the intrinsic objects behind usual (split) $2$-term ruths. We obtain a one-to-one correspondence between them, and relate to the well-known equivalence between $2$-term ruths and VB-groupoids/algebroids. On the other hand, we show that fat extensions of groupoids correspond to general linear PB-groupoids. The differentiation procedure of fat extensions is disc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain a one-to-one correspondence between [the category of fat extensions and the category of abstract 2-term representations up to homotopy], and relate to the well-known equivalence between 2-term ruths and VB-groupoids/algebroids. [...] we upgrade the one-to-one correspondence between general linear PB-groupoids and VB-groupoids of Cattafi and Garmendia to an equivalence of categories.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The newly defined categories of fat extensions and abstract 2-term ruths are well-posed and the stated one-to-one correspondences and equivalences hold under the standard assumptions of Lie groupoid theory without hidden restrictions on the objects involved.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Fat Lie theory defines fat extensions and abstract 2-term ruths with one-to-one correspondences to general linear PB-groupoids and core-transitive double groupoids, upgrading prior equivalences to category equivalences.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fat extensions of Lie groupoids correspond one-to-one with abstract 2-term representations up to homotopy.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"97591d687db9e1a2be0f0c391597f31931140c82129fd1f2504bd4b3f0361b16"},"source":{"id":"2603.08176","kind":"arxiv","version":2},"verdict":{"id":"37e064ee-d1ec-4a1f-ba6d-72914861ee55","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T14:07:54.599439Z","strongest_claim":"We obtain a one-to-one correspondence between [the category of fat extensions and the category of abstract 2-term representations up to homotopy], and relate to the well-known equivalence between 2-term ruths and VB-groupoids/algebroids. [...] we upgrade the one-to-one correspondence between general linear PB-groupoids and VB-groupoids of Cattafi and Garmendia to an equivalence of categories.","one_line_summary":"Fat Lie theory defines fat extensions and abstract 2-term ruths with one-to-one correspondences to general linear PB-groupoids and core-transitive double groupoids, upgrading prior equivalences to category equivalences.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The newly defined categories of fat extensions and abstract 2-term ruths are well-posed and the stated one-to-one correspondences and equivalences hold under the standard assumptions of Lie groupoid theory without hidden restrictions on the objects involved.","pith_extraction_headline":"Fat extensions of Lie groupoids correspond one-to-one with abstract 2-term representations up to homotopy."},"references":{"count":14,"sample":[{"doi":"","year":null,"title":"[CdH26] A. Cabrera and M. L. del Hoyo. Geometric differentiation of simplicial manifolds. Preprint, arXiv:2602.09885 [math.DG],","work_id":"4e91b61a-3041-41ac-8ada-170b3f7bbe56","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"[CF05] M. Crainic and R. L. Fernandes. Secondary characteristic classes of Lie algebroids. InQuantum field theory and noncommutative geometry. Based on the workshop, Sendai, Japan, November 2002, page","work_id":"2755bf44-4b46-4457-8b83-a9ffdaf28627","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"[CMS20] M. Crainic, J. N. Mestre, and I. Struchiner. Deformations of Lie groupoids.Int. Math. Res. Not., 2020(21):7662–7746,","work_id":"17b90ffe-9626-476b-9b35-5b1a61d7244c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"Carrillo Rouse","work_id":"3802f68c-ab11-4250-8fde-bd60bd6f1d16","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"arXiv preprint math/0403266 (2004)","work_id":"acc0217c-11e8-44cf-8fbd-efdfd632ac1b","ref_index":5,"cited_arxiv_id":"math/0403266","is_internal_anchor":true}],"resolved_work":14,"snapshot_sha256":"336accb0497aaf36f84575c7bb395a2310dbe70b2c83a14b22308dca47728f6f","internal_anchors":3},"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"}