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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. $L$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.03836","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-07-08T20:01:56Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"313377fdf6de0025bff3971f09fe3efcf81a9ea445c65e8a5056f9a746679da9","abstract_canon_sha256":"292c677d7beadb9ad85f566fe33abcb90f1b4c975d2dd42ab420db9015622b40"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:06.674056Z","signature_b64":"mLhpvDjWlpvihIHLxyDGwaewLD+yaL4tWrncimjsbMDJMdxjUjcoDplYsv13yzQ+DHJeMfzIgoUnamP6d9glBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a164055cf795d793557c9a119f68051923fe746b601e17feabc51d72888f490e","last_reissued_at":"2026-05-17T23:41:06.673336Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:06.673336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathcal{E}$. 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