{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:4CORVKLE2PZ4RT7IR4BTAF5ZBB","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":"0206df45f2b2008908afd8be3ee4818249e179a57b41beb9dec6cbc7b9d55659","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-06-30T09:24:20Z","title_canon_sha256":"24f3dc36e451da98c376d94967b51d69b0876287665b982743f799c60187322f"},"schema_version":"1.0","source":{"id":"2506.23651","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.23651","created_at":"2026-05-25T02:01:05Z"},{"alias_kind":"arxiv_version","alias_value":"2506.23651v2","created_at":"2026-05-25T02:01:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.23651","created_at":"2026-05-25T02:01:05Z"},{"alias_kind":"pith_short_12","alias_value":"4CORVKLE2PZ4","created_at":"2026-05-25T02:01:05Z"},{"alias_kind":"pith_short_16","alias_value":"4CORVKLE2PZ4RT7I","created_at":"2026-05-25T02:01:05Z"},{"alias_kind":"pith_short_8","alias_value":"4CORVKLE","created_at":"2026-05-25T02:01:05Z"}],"graph_snapshots":[{"event_id":"sha256:1ce59de8fa41885310ae49866b1f175dc5546e76e9e185aa3eb665dc3e04456e","target":"graph","created_at":"2026-05-25T02:01:05Z","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/2506.23651/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad -- a structure with 2-cells of all possible double-categorical shapes -- equipped with all possible composition operations, coherently. We also characterize them using \"implicit\" double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a \"ti","authors_text":"Aaron David Fairbanks, Michael Shulman","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-06-30T09:24:20Z","title":"Doubly weak double categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.23651","kind":"arxiv","version":2},"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:b3b5203d63276955c6e484e1f675b1a00b6d68886ff679547fcdaae0585582ba","target":"record","created_at":"2026-05-25T02:01:05Z","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":"0206df45f2b2008908afd8be3ee4818249e179a57b41beb9dec6cbc7b9d55659","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2025-06-30T09:24:20Z","title_canon_sha256":"24f3dc36e451da98c376d94967b51d69b0876287665b982743f799c60187322f"},"schema_version":"1.0","source":{"id":"2506.23651","kind":"arxiv","version":2}},"canonical_sha256":"e09d1aa964d3f3c8cfe88f033017b908757b2d31f284901973a6188c6ab1f10e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e09d1aa964d3f3c8cfe88f033017b908757b2d31f284901973a6188c6ab1f10e","first_computed_at":"2026-05-25T02:01:05.207180Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:01:05.207180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hzBlgFmrZf8XBI5KZnm1oRTKPvVJxm252KTfJySXQH+vMtxdsACE7MLBfqgePpMp6RLVbjgu+5x3zLhfs9JYCA==","signature_status":"signed_v1","signed_at":"2026-05-25T02:01:05.207857Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.23651","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b3b5203d63276955c6e484e1f675b1a00b6d68886ff679547fcdaae0585582ba","sha256:1ce59de8fa41885310ae49866b1f175dc5546e76e9e185aa3eb665dc3e04456e"],"state_sha256":"e0d509cff4f898e038c2028a855b3ec3a5995f3f3a7f2a0413c792e065b6fc6c"}