{"paper":{"title":"Birational Algebraic Topology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Dipankar Maity","submitted_at":"2026-06-22T05:58:08Z","abstract_excerpt":"Over a qcqs scheme $S$, we analyze the birational localization $L_{\\mathrm{bir}}\\mathcal{H}^{\\mathbb{A}^1}(S)$ of the motivic $\\infty$-category. As introduced in [\\cite{bachmann2019voevodsky}], this is obtained by localizing $\\mathcal{H}^{\\mathbb{A}^1}(S)$ at all dense open immersions in $Sm_S$. We establish that the associated localization functor $L_{bir}$ commutes with the bar construction, and thus preserves connectivity. Over a perfect field $k$, we demonstrate that a sheaf of groups is birational exactly when it is strongly $\\mathbb{A}^1$-invariant and has trivial $\\mathbb{G}_m$-contract"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22887","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/2606.22887/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"}