{"paper":{"title":"Localization theory in an $\\infty$-topos","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Marco Vergura","submitted_at":"2019-07-08T20:01:56Z","abstract_excerpt":"We develop the theory of reflective subfibrations on an $\\infty$-topos $\\mathcal{E}$. A reflective subfibration $L_\\bullet$ on $\\mathcal{E}$ is a pullback-compatible assignment of a reflective subcategory $\\mathcal{D}_X\\subseteq \\mathcal{E}{/X}$, for every $X \\in \\mathcal{E}$. Reflective subfibrations abound in homotopy theory, albeit often disguised, e.g., as stable factorization systems. We prove that $L$-local maps (i.e., those maps that belong to some $\\mathcal{D}_X$) admit a classifying map, and we introduce the class of $L$-separated maps, that is, those maps with $L$-local diagonal. $L$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03836","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":""},"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"}