{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:VHSDHBEYEVLJFXIHYVKFXVNF5L","short_pith_number":"pith:VHSDHBEY","canonical_record":{"source":{"id":"1803.03418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-03-09T08:53:41Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"236d96270533b11bff3518c1b0ba16ecdf7be640483b80b779650761851df584","abstract_canon_sha256":"dd874669f66fa072a59cf32a30cb02c85e7c34a9c9bf1673f461f4503f4ba9a9"},"schema_version":"1.0"},"canonical_sha256":"a9e4338498255692dd07c5545bd5a5ead5d701cf457c3e543ef7319f8af685ba","source":{"kind":"arxiv","id":"1803.03418","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.03418","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"arxiv_version","alias_value":"1803.03418v1","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03418","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"pith_short_12","alias_value":"VHSDHBEYEVLJ","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VHSDHBEYEVLJFXIH","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VHSDHBEY","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:VHSDHBEYEVLJFXIHYVKFXVNF5L","target":"record","payload":{"canonical_record":{"source":{"id":"1803.03418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-03-09T08:53:41Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"236d96270533b11bff3518c1b0ba16ecdf7be640483b80b779650761851df584","abstract_canon_sha256":"dd874669f66fa072a59cf32a30cb02c85e7c34a9c9bf1673f461f4503f4ba9a9"},"schema_version":"1.0"},"canonical_sha256":"a9e4338498255692dd07c5545bd5a5ead5d701cf457c3e543ef7319f8af685ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:39.655768Z","signature_b64":"s59luBauqtkvIq0YVzNM+suMtYIhfyKIiUL4VyCPI3jyKW+jCuXrav+QoFYgsT49mjNykS3O942AmkjcZb/KCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9e4338498255692dd07c5545bd5a5ead5d701cf457c3e543ef7319f8af685ba","last_reissued_at":"2026-05-18T00:21:39.655073Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:39.655073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.03418","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:21:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U0/H5gF3zvON9CGIIECAG1DgIXqkt0Wu8T3FYzDHxXky2o78pH0IHK7MBPqCaFA0ekWdsz4n/l2eOxwY5rHYAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T11:51:16.338954Z"},"content_sha256":"da0bead5403b74c6f2e5eb8428c67e39d95e67bc451c4083440a9dfc7257bc0d","schema_version":"1.0","event_id":"sha256:da0bead5403b74c6f2e5eb8428c67e39d95e67bc451c4083440a9dfc7257bc0d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:VHSDHBEYEVLJFXIHYVKFXVNF5L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Crossed modules of monoids I. Relative categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.CT","authors_text":"Gabriella B\\\"ohm","submitted_at":"2018-03-09T08:53:41Z","abstract_excerpt":"This is the first part of a series of three strongly related papers in which three equivalent structures are studied:\n  - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans\n  - crossed modules of monoids relative to this class of spans\n  - simplicial monoids of so-called Moore length 1 relative to this class of spans.\n  The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03418","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T00:21:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0sH+ee/dxbdd5S99D1QBFHji9KA3kE+cIXa6a6sug9RqIA+Vgx/2i+0YBwt2hXuhe/8ggGeBvoFHXSh+/OBJDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T11:51:16.339296Z"},"content_sha256":"b965dde34a5e5e2b84a009b0a35600f03cc1341ffa7af446de3f69e00b018b2e","schema_version":"1.0","event_id":"sha256:b965dde34a5e5e2b84a009b0a35600f03cc1341ffa7af446de3f69e00b018b2e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VHSDHBEYEVLJFXIHYVKFXVNF5L/bundle.json","state_url":"https://pith.science/pith/VHSDHBEYEVLJFXIHYVKFXVNF5L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VHSDHBEYEVLJFXIHYVKFXVNF5L/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":"2026-06-25T11:51:16Z","links":{"resolver":"https://pith.science/pith/VHSDHBEYEVLJFXIHYVKFXVNF5L","bundle":"https://pith.science/pith/VHSDHBEYEVLJFXIHYVKFXVNF5L/bundle.json","state":"https://pith.science/pith/VHSDHBEYEVLJFXIHYVKFXVNF5L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VHSDHBEYEVLJFXIHYVKFXVNF5L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VHSDHBEYEVLJFXIHYVKFXVNF5L","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":"dd874669f66fa072a59cf32a30cb02c85e7c34a9c9bf1673f461f4503f4ba9a9","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-03-09T08:53:41Z","title_canon_sha256":"236d96270533b11bff3518c1b0ba16ecdf7be640483b80b779650761851df584"},"schema_version":"1.0","source":{"id":"1803.03418","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.03418","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"arxiv_version","alias_value":"1803.03418v1","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03418","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"pith_short_12","alias_value":"VHSDHBEYEVLJ","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VHSDHBEYEVLJFXIH","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VHSDHBEY","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:b965dde34a5e5e2b84a009b0a35600f03cc1341ffa7af446de3f69e00b018b2e","target":"graph","created_at":"2026-05-18T00:21:39Z","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"},"paper":{"abstract_excerpt":"This is the first part of a series of three strongly related papers in which three equivalent structures are studied:\n  - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans\n  - crossed modules of monoids relative to this class of spans\n  - simplicial monoids of so-called Moore length 1 relative to this class of spans.\n  The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids).","authors_text":"Gabriella B\\\"ohm","cross_cats":["math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-03-09T08:53:41Z","title":"Crossed modules of monoids I. Relative categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03418","kind":"arxiv","version":1},"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:da0bead5403b74c6f2e5eb8428c67e39d95e67bc451c4083440a9dfc7257bc0d","target":"record","created_at":"2026-05-18T00:21:39Z","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":"dd874669f66fa072a59cf32a30cb02c85e7c34a9c9bf1673f461f4503f4ba9a9","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-03-09T08:53:41Z","title_canon_sha256":"236d96270533b11bff3518c1b0ba16ecdf7be640483b80b779650761851df584"},"schema_version":"1.0","source":{"id":"1803.03418","kind":"arxiv","version":1}},"canonical_sha256":"a9e4338498255692dd07c5545bd5a5ead5d701cf457c3e543ef7319f8af685ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9e4338498255692dd07c5545bd5a5ead5d701cf457c3e543ef7319f8af685ba","first_computed_at":"2026-05-18T00:21:39.655073Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:39.655073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s59luBauqtkvIq0YVzNM+suMtYIhfyKIiUL4VyCPI3jyKW+jCuXrav+QoFYgsT49mjNykS3O942AmkjcZb/KCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:39.655768Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.03418","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da0bead5403b74c6f2e5eb8428c67e39d95e67bc451c4083440a9dfc7257bc0d","sha256:b965dde34a5e5e2b84a009b0a35600f03cc1341ffa7af446de3f69e00b018b2e"],"state_sha256":"445f2bc40035617758f137f210b56f9278b8b0afa38dbf1fc601c081e2bd3e6a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rCX0uBIfpY/LZPZNq5WKwlxa2m/aAWWFBMkNJS5pNPlxoyiaWnDLQkKezxwykBZN/rw9Apeg69+l4MZaBwwyAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T11:51:16.341223Z","bundle_sha256":"39c7a9cb97ec104a18bd54203e07a997c7b5fffb2b6b2e3faeea6c1e7ad194e0"}}