{"paper":{"title":"Constructions of Waldhausen categories via Grothendieck opfibrations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RT","authors_text":"Li Liang, Liping Li, Zhenxing Di","submitted_at":"2024-07-22T13:06:10Z","abstract_excerpt":"Given a Grothendieck opfibration $p: \\mathcal{T} \\to \\mathcal{B}$, we describe a method to construct a Waldhausen category structure on the total category $\\mathcal{T}$ via combining Waldhausen category structures on the fibers $\\mathcal{T}_A$ for $A \\in \\mathrm{Ob}(\\mathcal{B})$ and the basis category $\\mathcal{B}$. As an application, we show that if $\\mathsf{E}$ is a Waldhausen category with small coproducts such that the class of cofibrations is the left part of a weak factorization system in $\\mathsf{E}$, then the representation category $\\mathsf{Rep}(Q, \\mathsf{coE})$ of a left rooted qui"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.15607","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.15607/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}