{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:XLGC5Z5HJ4MF5JQVHEWZM7I56Z","short_pith_number":"pith:XLGC5Z5H","canonical_record":{"source":{"id":"2504.11099","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-04-15T11:45:59Z","cross_cats_sorted":[],"title_canon_sha256":"95f1f76239b4b8261d9efda7db177386df65a179008c1029af2dd322497a5159","abstract_canon_sha256":"8e95bdcb30962d8ade48e391159603dbc3182f4909ca62de52e3de098769d725"},"schema_version":"1.0"},"canonical_sha256":"bacc2ee7a74f185ea615392d967d1df65a611493558cfaef4c85f4f2a1efbf7f","source":{"kind":"arxiv","id":"2504.11099","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.11099","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"arxiv_version","alias_value":"2504.11099v4","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.11099","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"pith_short_12","alias_value":"XLGC5Z5HJ4MF","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"pith_short_16","alias_value":"XLGC5Z5HJ4MF5JQV","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"pith_short_8","alias_value":"XLGC5Z5H","created_at":"2026-06-10T01:09:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:XLGC5Z5HJ4MF5JQVHEWZM7I56Z","target":"record","payload":{"canonical_record":{"source":{"id":"2504.11099","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-04-15T11:45:59Z","cross_cats_sorted":[],"title_canon_sha256":"95f1f76239b4b8261d9efda7db177386df65a179008c1029af2dd322497a5159","abstract_canon_sha256":"8e95bdcb30962d8ade48e391159603dbc3182f4909ca62de52e3de098769d725"},"schema_version":"1.0"},"canonical_sha256":"bacc2ee7a74f185ea615392d967d1df65a611493558cfaef4c85f4f2a1efbf7f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:09:16.265786Z","signature_b64":"BYesdeNYcT+0SpjtzNtprUrEpldvhlpM+nvcmWo0Cdzs37pVbphyW7tsuqj58b9Gl769MhawLYIMaDKC/Qa+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bacc2ee7a74f185ea615392d967d1df65a611493558cfaef4c85f4f2a1efbf7f","last_reissued_at":"2026-06-10T01:09:16.265272Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:09:16.265272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2504.11099","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":"2026-06-10T01:09:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IPOgr2zbndIyQRFG1ESEM0e4UV7YzJn2GAzPTGorQpXtRc/tzjnBW9LxChBA+cNXKKUfWDRiL+2lWRdz9DGpAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T09:29:16.839107Z"},"content_sha256":"7d92e761ad4d4793ba4232808d25da03638cabea4e475d31354b611ea44a8aef","schema_version":"1.0","event_id":"sha256:7d92e761ad4d4793ba4232808d25da03638cabea4e475d31354b611ea44a8aef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:XLGC5Z5HJ4MF5JQVHEWZM7I56Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Double categories of profunctors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Virtual double categories of enriched categories, functors, and profunctors are characterized by a new notion of double-categorical colimits.","cross_cats":[],"primary_cat":"math.CT","authors_text":"Yuto Kawase","submitted_at":"2025-04-15T11:45:59Z","abstract_excerpt":"We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual double categories and, at the level of objects, up to isomorphism of enriched categories. Throughout the paper, we treat enrichment in a unital virtual double category rather than in a bicategory or a monoidal category, and, for consistency and better visualization of pasting diagrams, we adopt augmented virtual double categories as a fundamental language f"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual double categories and, at the level of objects, up to isomorphism of enriched categories.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The choice to treat enrichment in a unital virtual double category rather than in a bicategory or a monoidal category, together with the adoption of augmented virtual double categories as the fundamental language for double-categorical concepts.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Characterizes virtual double categories of enriched categories, functors, and profunctors via a new notion of double-categorical colimits, strictly up to equivalence.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Virtual double categories of enriched categories, functors, and profunctors are characterized by a new notion of double-categorical colimits.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f97e850d8ff3597e570597b0b405453909b331030d917e2c037b9c99949e2739"},"source":{"id":"2504.11099","kind":"arxiv","version":4},"verdict":{"id":"fc5b2eee-6c7c-4dea-b3c3-04dc3e858920","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-22T21:03:34.192550Z","strongest_claim":"We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual double categories and, at the level of objects, up to isomorphism of enriched categories.","one_line_summary":"Characterizes virtual double categories of enriched categories, functors, and profunctors via a new notion of double-categorical colimits, strictly up to equivalence.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The choice to treat enrichment in a unital virtual double category rather than in a bicategory or a monoidal category, together with the adoption of augmented virtual double categories as the fundamental language for double-categorical concepts.","pith_extraction_headline":"Virtual double categories of enriched categories, functors, and profunctors are characterized by a new notion of double-categorical colimits."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.11099/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":"fc5b2eee-6c7c-4dea-b3c3-04dc3e858920"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-10T01:09:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HnLk9dq3MzfvwZouv8KVYydSG55gZTGQjgrvYmXVnRiKQM1bJRs2pqyXz6T/0WszetedV3GJjBLZCvU+1VXlBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T09:29:16.839778Z"},"content_sha256":"18b49084bf44eb326d31d67f09d35b4c31c56acaed8775e138fee1c1610208c6","schema_version":"1.0","event_id":"sha256:18b49084bf44eb326d31d67f09d35b4c31c56acaed8775e138fee1c1610208c6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XLGC5Z5HJ4MF5JQVHEWZM7I56Z/bundle.json","state_url":"https://pith.science/pith/XLGC5Z5HJ4MF5JQVHEWZM7I56Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XLGC5Z5HJ4MF5JQVHEWZM7I56Z/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":"2026-08-07T09:29:16Z","links":{"resolver":"https://pith.science/pith/XLGC5Z5HJ4MF5JQVHEWZM7I56Z","bundle":"https://pith.science/pith/XLGC5Z5HJ4MF5JQVHEWZM7I56Z/bundle.json","state":"https://pith.science/pith/XLGC5Z5HJ4MF5JQVHEWZM7I56Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XLGC5Z5HJ4MF5JQVHEWZM7I56Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:XLGC5Z5HJ4MF5JQVHEWZM7I56Z","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":"8e95bdcb30962d8ade48e391159603dbc3182f4909ca62de52e3de098769d725","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-04-15T11:45:59Z","title_canon_sha256":"95f1f76239b4b8261d9efda7db177386df65a179008c1029af2dd322497a5159"},"schema_version":"1.0","source":{"id":"2504.11099","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.11099","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"arxiv_version","alias_value":"2504.11099v4","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.11099","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"pith_short_12","alias_value":"XLGC5Z5HJ4MF","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"pith_short_16","alias_value":"XLGC5Z5HJ4MF5JQV","created_at":"2026-06-10T01:09:16Z"},{"alias_kind":"pith_short_8","alias_value":"XLGC5Z5H","created_at":"2026-06-10T01:09:16Z"}],"graph_snapshots":[{"event_id":"sha256:18b49084bf44eb326d31d67f09d35b4c31c56acaed8775e138fee1c1610208c6","target":"graph","created_at":"2026-06-10T01:09:16Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual double categories and, at the level of objects, up to isomorphism of enriched categories."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The choice to treat enrichment in a unital virtual double category rather than in a bicategory or a monoidal category, together with the adoption of augmented virtual double categories as the fundamental language for double-categorical concepts."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Characterizes virtual double categories of enriched categories, functors, and profunctors via a new notion of double-categorical colimits, strictly up to equivalence."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Virtual double categories of enriched categories, functors, and profunctors are characterized by a new notion of double-categorical colimits."}],"snapshot_sha256":"f97e850d8ff3597e570597b0b405453909b331030d917e2c037b9c99949e2739"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2504.11099/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual double categories and, at the level of objects, up to isomorphism of enriched categories. Throughout the paper, we treat enrichment in a unital virtual double category rather than in a bicategory or a monoidal category, and, for consistency and better visualization of pasting diagrams, we adopt augmented virtual double categories as a fundamental language f","authors_text":"Yuto Kawase","cross_cats":[],"headline":"Virtual double categories of enriched categories, functors, and profunctors are characterized by a new notion of double-categorical colimits.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-04-15T11:45:59Z","title":"Double categories of profunctors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.11099","kind":"arxiv","version":4},"verdict":{"created_at":"2026-05-22T21:03:34.192550Z","id":"fc5b2eee-6c7c-4dea-b3c3-04dc3e858920","model_set":{"reader":"grok-4.3"},"one_line_summary":"Characterizes virtual double categories of enriched categories, functors, and profunctors via a new notion of double-categorical colimits, strictly up to equivalence.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Virtual double categories of enriched categories, functors, and profunctors are characterized by a new notion of double-categorical colimits.","strongest_claim":"We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual double categories and, at the level of objects, up to isomorphism of enriched categories.","weakest_assumption":"The choice to treat enrichment in a unital virtual double category rather than in a bicategory or a monoidal category, together with the adoption of augmented virtual double categories as the fundamental language for double-categorical concepts."}},"verdict_id":"fc5b2eee-6c7c-4dea-b3c3-04dc3e858920"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7d92e761ad4d4793ba4232808d25da03638cabea4e475d31354b611ea44a8aef","target":"record","created_at":"2026-06-10T01:09:16Z","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":"8e95bdcb30962d8ade48e391159603dbc3182f4909ca62de52e3de098769d725","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-04-15T11:45:59Z","title_canon_sha256":"95f1f76239b4b8261d9efda7db177386df65a179008c1029af2dd322497a5159"},"schema_version":"1.0","source":{"id":"2504.11099","kind":"arxiv","version":4}},"canonical_sha256":"bacc2ee7a74f185ea615392d967d1df65a611493558cfaef4c85f4f2a1efbf7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bacc2ee7a74f185ea615392d967d1df65a611493558cfaef4c85f4f2a1efbf7f","first_computed_at":"2026-06-10T01:09:16.265272Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:09:16.265272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BYesdeNYcT+0SpjtzNtprUrEpldvhlpM+nvcmWo0Cdzs37pVbphyW7tsuqj58b9Gl769MhawLYIMaDKC/Qa+Cg==","signature_status":"signed_v1","signed_at":"2026-06-10T01:09:16.265786Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.11099","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7d92e761ad4d4793ba4232808d25da03638cabea4e475d31354b611ea44a8aef","sha256:18b49084bf44eb326d31d67f09d35b4c31c56acaed8775e138fee1c1610208c6"],"state_sha256":"df900eedb5ebfcbbe596baba9ad9a8c13ae70be173f619879e141152520f9fce"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g7zR7lOCQU+XVgez4SMt1lABwG7DpJUceq9+gttFJ1q7QM1pcBuk9Ktxc7hOc+pTOfGBaoqV7mkQA3qHjdZAAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T09:29:16.844127Z","bundle_sha256":"ec64e744a4236bcb3985bda88abfa9c9187cd7bf911b237e91a2cb5daa3eafaf"}}