{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:3523PUTG55A2K7K3RT7C4GPAJU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1a9f2bea023e7129c1dca1e734873aaafbbbfccfed8df28e8d0122c7d9e2de83","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2022-10-21T10:31:44Z","title_canon_sha256":"93ea657b83abac261b7d0711fee4d6037162c0859921f10eaa43aa72a746c069"},"schema_version":"1.0","source":{"id":"2210.11861","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.11861","created_at":"2026-07-30T01:18:35Z"},{"alias_kind":"arxiv_version","alias_value":"2210.11861v4","created_at":"2026-07-30T01:18:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.11861","created_at":"2026-07-30T01:18:35Z"},{"alias_kind":"pith_short_12","alias_value":"3523PUTG55A2","created_at":"2026-07-30T01:18:35Z"},{"alias_kind":"pith_short_16","alias_value":"3523PUTG55A2K7K3","created_at":"2026-07-30T01:18:35Z"},{"alias_kind":"pith_short_8","alias_value":"3523PUTG","created_at":"2026-07-30T01:18:35Z"}],"graph_snapshots":[{"event_id":"sha256:d4e510dbed45a77535be598dd4cb30aeed481d232c4d26678571fbf3f2005abf","target":"graph","created_at":"2026-07-30T01:18:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2210.11861/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This semi-expository work covers central aspects of the theory of relative tensor products as developed in Higher Algebra, as well as their application to Koszul duality for algebras in monoidal oo-categories. Part of our goal is to expand on the rather condensed account of loc. cit. Along the way, we generalize various aspects of the theory. For instance, given a monoidal oo-category Cc, an oo-category Mm which is left-tensored over Cc, and an algebra A in Cc, we construct an action of A-A-bimodules N in Cc on left A-modules M in Mm by an \"external relative tensor product\" N \\otimes_A M. (Up ","authors_text":"Asaf Horev, Ishai Dan-Cohen","cross_cats":["math.AT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2022-10-21T10:31:44Z","title":"Relative tensor products and Koszul duality in monoidal oo-categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.11861","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d58e709812239ddf64d48a56a4d3ac12ea197fafe60c3ce45b21a600f58bbb31","target":"record","created_at":"2026-07-30T01:18:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1a9f2bea023e7129c1dca1e734873aaafbbbfccfed8df28e8d0122c7d9e2de83","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2022-10-21T10:31:44Z","title_canon_sha256":"93ea657b83abac261b7d0711fee4d6037162c0859921f10eaa43aa72a746c069"},"schema_version":"1.0","source":{"id":"2210.11861","kind":"arxiv","version":4}},"canonical_sha256":"df75b7d266ef41a57d5b8cfe2e19e04d0733a9cc0f6d6006e84fd14334211318","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"df75b7d266ef41a57d5b8cfe2e19e04d0733a9cc0f6d6006e84fd14334211318","first_computed_at":"2026-07-30T01:18:35.388836Z","kind":"pith_receipt","last_reissued_at":"2026-07-30T01:18:35.388836Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2210.11861","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d58e709812239ddf64d48a56a4d3ac12ea197fafe60c3ce45b21a600f58bbb31","sha256:d4e510dbed45a77535be598dd4cb30aeed481d232c4d26678571fbf3f2005abf"],"state_sha256":"378f4608be1ecaaa97ab05ec09f250c46c127e7fdf853ccdd5b640d24f2abeda"}