{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:3523PUTG55A2K7K3RT7C4GPAJU","short_pith_number":"pith:3523PUTG","canonical_record":{"source":{"id":"2210.11861","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2022-10-21T10:31:44Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"93ea657b83abac261b7d0711fee4d6037162c0859921f10eaa43aa72a746c069","abstract_canon_sha256":"1a9f2bea023e7129c1dca1e734873aaafbbbfccfed8df28e8d0122c7d9e2de83"},"schema_version":"1.0"},"canonical_sha256":"df75b7d266ef41a57d5b8cfe2e19e04d0733a9cc0f6d6006e84fd14334211318","source":{"kind":"arxiv","id":"2210.11861","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:3523PUTG55A2K7K3RT7C4GPAJU","target":"record","payload":{"canonical_record":{"source":{"id":"2210.11861","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2022-10-21T10:31:44Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"93ea657b83abac261b7d0711fee4d6037162c0859921f10eaa43aa72a746c069","abstract_canon_sha256":"1a9f2bea023e7129c1dca1e734873aaafbbbfccfed8df28e8d0122c7d9e2de83"},"schema_version":"1.0"},"canonical_sha256":"df75b7d266ef41a57d5b8cfe2e19e04d0733a9cc0f6d6006e84fd14334211318","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df75b7d266ef41a57d5b8cfe2e19e04d0733a9cc0f6d6006e84fd14334211318","last_reissued_at":"2026-07-30T01:18:35.388836Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:18:35.388836Z"},"source_kind":"arxiv","source_id":"2210.11861","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-30T01:18:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lh/tWw/kFv/1auz97m/ACKKVAxVVv7JMvC4/fMsgHM3flkH3nZPyNMgW/JTeAuCah3aGrWP/ibtsnzYj/bUfDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T19:41:49.279603Z"},"content_sha256":"d58e709812239ddf64d48a56a4d3ac12ea197fafe60c3ce45b21a600f58bbb31","schema_version":"1.0","event_id":"sha256:d58e709812239ddf64d48a56a4d3ac12ea197fafe60c3ce45b21a600f58bbb31"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:3523PUTG55A2K7K3RT7C4GPAJU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Relative tensor products and Koszul duality in monoidal oo-categories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.AG","authors_text":"Asaf Horev, Ishai Dan-Cohen","submitted_at":"2022-10-21T10:31:44Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.11861","kind":"arxiv","version":4},"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/2210.11861/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-30T01:18:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RYy2XcukUv5JcjwLT64mAvYXHpSi0pZ9LFRk8ar4k/Zz9HHl8iWgR/dfRm72eLmvmLyzON7k5/eRUbQzhISjAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T19:41:49.280626Z"},"content_sha256":"d4e510dbed45a77535be598dd4cb30aeed481d232c4d26678571fbf3f2005abf","schema_version":"1.0","event_id":"sha256:d4e510dbed45a77535be598dd4cb30aeed481d232c4d26678571fbf3f2005abf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3523PUTG55A2K7K3RT7C4GPAJU/bundle.json","state_url":"https://pith.science/pith/3523PUTG55A2K7K3RT7C4GPAJU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3523PUTG55A2K7K3RT7C4GPAJU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-21T19:41:49Z","links":{"resolver":"https://pith.science/pith/3523PUTG55A2K7K3RT7C4GPAJU","bundle":"https://pith.science/pith/3523PUTG55A2K7K3RT7C4GPAJU/bundle.json","state":"https://pith.science/pith/3523PUTG55A2K7K3RT7C4GPAJU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3523PUTG55A2K7K3RT7C4GPAJU/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3t/EJX8HIa6290JmVI4+NCqAY2hDTmgMssOcvIRPBF9/c4lKVWg5R4KGkLJ1Rs6VtYEnUewGPLIpFsh2TJ4ACg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T19:41:49.294182Z","bundle_sha256":"5c5e7e93c0bf2b843a6b75a2da9eaa3413bb04c833eec6abf8c36daeeb7e7dd5"}}