{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:4OTB6HN7TVZI5KPE54KQ53UAQ4","short_pith_number":"pith:4OTB6HN7","canonical_record":{"source":{"id":"math/0510490","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2005-10-23T19:26:06Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"32b114728aa429328fe251f0a089b362450f28e3bbb70c4f35f79090850f8a78","abstract_canon_sha256":"6ef72eb80a9b5f15b5e8d5ca477c487f2d1d68b4837e7fb661e6fce25db118df"},"schema_version":"1.0"},"canonical_sha256":"e3a61f1dbf9d728ea9e4ef150eee8087345862c93456ba59901f010f82519e26","source":{"kind":"arxiv","id":"math/0510490","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0510490","created_at":"2026-05-18T03:58:38Z"},{"alias_kind":"arxiv_version","alias_value":"math/0510490v2","created_at":"2026-05-18T03:58:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0510490","created_at":"2026-05-18T03:58:38Z"},{"alias_kind":"pith_short_12","alias_value":"4OTB6HN7TVZI","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"4OTB6HN7TVZI5KPE","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"4OTB6HN7","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:4OTB6HN7TVZI5KPE54KQ53UAQ4","target":"record","payload":{"canonical_record":{"source":{"id":"math/0510490","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2005-10-23T19:26:06Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"32b114728aa429328fe251f0a089b362450f28e3bbb70c4f35f79090850f8a78","abstract_canon_sha256":"6ef72eb80a9b5f15b5e8d5ca477c487f2d1d68b4837e7fb661e6fce25db118df"},"schema_version":"1.0"},"canonical_sha256":"e3a61f1dbf9d728ea9e4ef150eee8087345862c93456ba59901f010f82519e26","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:38.547393Z","signature_b64":"y8oupHTFk/XYtbIPvRxO7kN3TnXefQ6C8ygZctIVAn6ic3kMMc//8ZgAF8wLW/v7JqnsH9+DSU/vt18GTracCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3a61f1dbf9d728ea9e4ef150eee8087345862c93456ba59901f010f82519e26","last_reissued_at":"2026-05-18T03:58:38.546907Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:38.546907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0510490","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:58:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y0o/zXLIyF2vvbt4wnv1KAEdhqWFyXxdW9iyuSHG0Oz6CmnKbQWE2/1jqE/w7p8/kSVmi7EiPyS0Ko0QQ3U/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:44:50.105008Z"},"content_sha256":"cc123708465440cfe13d08d8e2a5845a2563c19665c8598d3ac7bd6d324f8f54","schema_version":"1.0","event_id":"sha256:cc123708465440cfe13d08d8e2a5845a2563c19665c8598d3ac7bd6d324f8f54"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:4OTB6HN7TVZI5KPE54KQ53UAQ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on the composition product of symmetric sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CT","authors_text":"Michael Ching","submitted_at":"2005-10-23T19:26:06Z","abstract_excerpt":"We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough structure, namely that of a `normal oplax' monoidal product, to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit of this work is in the dual setting, where it allows us to define a cobar construction for cooperads."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0510490","kind":"arxiv","version":2},"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":"2026-05-18T03:58:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nN+pCqjSTOUYUMAPPVl+nO349GlwWXk0lGnftVi3Sd5SfJ9Tj8E8MuKkk11CWQ2krjHCXBTXMtUMv3qhlwOGAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:44:50.105386Z"},"content_sha256":"63401b3ac30840a44cf9b5317f8378787906c013b54a5d5c671fbe425af5ccb1","schema_version":"1.0","event_id":"sha256:63401b3ac30840a44cf9b5317f8378787906c013b54a5d5c671fbe425af5ccb1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4OTB6HN7TVZI5KPE54KQ53UAQ4/bundle.json","state_url":"https://pith.science/pith/4OTB6HN7TVZI5KPE54KQ53UAQ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4OTB6HN7TVZI5KPE54KQ53UAQ4/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":"2026-06-24T22:44:50Z","links":{"resolver":"https://pith.science/pith/4OTB6HN7TVZI5KPE54KQ53UAQ4","bundle":"https://pith.science/pith/4OTB6HN7TVZI5KPE54KQ53UAQ4/bundle.json","state":"https://pith.science/pith/4OTB6HN7TVZI5KPE54KQ53UAQ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4OTB6HN7TVZI5KPE54KQ53UAQ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:4OTB6HN7TVZI5KPE54KQ53UAQ4","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":"6ef72eb80a9b5f15b5e8d5ca477c487f2d1d68b4837e7fb661e6fce25db118df","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2005-10-23T19:26:06Z","title_canon_sha256":"32b114728aa429328fe251f0a089b362450f28e3bbb70c4f35f79090850f8a78"},"schema_version":"1.0","source":{"id":"math/0510490","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0510490","created_at":"2026-05-18T03:58:38Z"},{"alias_kind":"arxiv_version","alias_value":"math/0510490v2","created_at":"2026-05-18T03:58:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0510490","created_at":"2026-05-18T03:58:38Z"},{"alias_kind":"pith_short_12","alias_value":"4OTB6HN7TVZI","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"4OTB6HN7TVZI5KPE","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"4OTB6HN7","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:63401b3ac30840a44cf9b5317f8378787906c013b54a5d5c671fbe425af5ccb1","target":"graph","created_at":"2026-05-18T03:58:38Z","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":"We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough structure, namely that of a `normal oplax' monoidal product, to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit of this work is in the dual setting, where it allows us to define a cobar construction for cooperads.","authors_text":"Michael Ching","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2005-10-23T19:26:06Z","title":"A note on the composition product of symmetric sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0510490","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:cc123708465440cfe13d08d8e2a5845a2563c19665c8598d3ac7bd6d324f8f54","target":"record","created_at":"2026-05-18T03:58:38Z","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":"6ef72eb80a9b5f15b5e8d5ca477c487f2d1d68b4837e7fb661e6fce25db118df","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2005-10-23T19:26:06Z","title_canon_sha256":"32b114728aa429328fe251f0a089b362450f28e3bbb70c4f35f79090850f8a78"},"schema_version":"1.0","source":{"id":"math/0510490","kind":"arxiv","version":2}},"canonical_sha256":"e3a61f1dbf9d728ea9e4ef150eee8087345862c93456ba59901f010f82519e26","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3a61f1dbf9d728ea9e4ef150eee8087345862c93456ba59901f010f82519e26","first_computed_at":"2026-05-18T03:58:38.546907Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:38.546907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y8oupHTFk/XYtbIPvRxO7kN3TnXefQ6C8ygZctIVAn6ic3kMMc//8ZgAF8wLW/v7JqnsH9+DSU/vt18GTracCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:38.547393Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0510490","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cc123708465440cfe13d08d8e2a5845a2563c19665c8598d3ac7bd6d324f8f54","sha256:63401b3ac30840a44cf9b5317f8378787906c013b54a5d5c671fbe425af5ccb1"],"state_sha256":"3eb15b72ab50b827ca578325ed33719fb2dd584eff3db839247fa229311eb056"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V2Mg54MPdeLXhj800aHDRIuldA/ef7HSLWoSQF7SITCBmVbpEaSsoQf61N3viM5YzNTjmNV9jkWCPt4vr/cHDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T22:44:50.107475Z","bundle_sha256":"ef280d83a20f7af5e6d247791a882f28d9238c8ec01d6a0b9e7a59468d9ea62d"}}