{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:EQ6464QR7V26OGER2E27YY6YPZ","short_pith_number":"pith:EQ6464QR","canonical_record":{"source":{"id":"1601.04123","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-16T04:51:11Z","cross_cats_sorted":[],"title_canon_sha256":"a702214f6da8cb3837ad7528ae427626e9173d141f91f35e54d89504bae345c2","abstract_canon_sha256":"83fedc8cbfa1417987fd37d3c1cb089ab30d0ebb20f4d724f038c7f12fe94b99"},"schema_version":"1.0"},"canonical_sha256":"243dcf7211fd75e71891d135fc63d87e40570558439cd3702a5bc1dbc68df9b4","source":{"kind":"arxiv","id":"1601.04123","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.04123","created_at":"2026-05-18T00:44:58Z"},{"alias_kind":"arxiv_version","alias_value":"1601.04123v4","created_at":"2026-05-18T00:44:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04123","created_at":"2026-05-18T00:44:58Z"},{"alias_kind":"pith_short_12","alias_value":"EQ6464QR7V26","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"EQ6464QR7V26OGER","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"EQ6464QR","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:EQ6464QR7V26OGER2E27YY6YPZ","target":"record","payload":{"canonical_record":{"source":{"id":"1601.04123","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-16T04:51:11Z","cross_cats_sorted":[],"title_canon_sha256":"a702214f6da8cb3837ad7528ae427626e9173d141f91f35e54d89504bae345c2","abstract_canon_sha256":"83fedc8cbfa1417987fd37d3c1cb089ab30d0ebb20f4d724f038c7f12fe94b99"},"schema_version":"1.0"},"canonical_sha256":"243dcf7211fd75e71891d135fc63d87e40570558439cd3702a5bc1dbc68df9b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:58.195836Z","signature_b64":"ox0lgDgJDrdHrcuJise1d2bMwLdMgqzxvIm96Ij81xwGNop7snzfT6FUofUa6E4MdsNFuAwzSCmHydJtlfbXAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"243dcf7211fd75e71891d135fc63d87e40570558439cd3702a5bc1dbc68df9b4","last_reissued_at":"2026-05-18T00:44:58.195392Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:58.195392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.04123","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":"2026-05-18T00:44:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kalLEq+HyB0m7yiC6P+QnACg651YUIAX6mYcLeyN4uyPAehXPS/CQ6Hfc29sdV38goI8gp8NsbeSTacb3uoQBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T00:32:37.274594Z"},"content_sha256":"7371eddb6c1dab5dbe655576b6717611a357b2e8cb8f39088e2df86b9c6bc8dc","schema_version":"1.0","event_id":"sha256:7371eddb6c1dab5dbe655576b6717611a357b2e8cb8f39088e2df86b9c6bc8dc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:EQ6464QR7V26OGER2E27YY6YPZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Relative Thom Spectra Via Operadic Kan Extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Jonathan Beardsley","submitted_at":"2016-01-16T04:51:11Z","abstract_excerpt":"We show that a large number of Thom spectra, i.e. colimits of morphisms $BG\\to BGL_1(\\mathbb{S})$, can be obtained as iterated Thom spectra, i.e. colimits of morphisms $BG\\to BGL_1(Mf)$ for some Thom spectrum $Mf$. This leads to a number of new relative Thom isomorphisms, e.g. $MU[6,\\infty)\\wedge_{MString} MU[6,\\infty)\\simeq MU[6,\\infty)\\wedge\\mathbb{S}[B^3Spin]$. As an example of interest to chromatic homotopy theorists, we also show that Ravenel's $X(n)$ filtration of $MU$ is a tower of intermediate Thom spectra determined by a natural filtration of $BU$ by sub-bialagebras."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04123","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":""},"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":"2026-05-18T00:44:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GngWhhZfPlZHvtluwqRgr1IgVlUhQZ+5e1flTWQsAvRsrFojDOAWEGlG0raOeTXe26242Nqi8d0zsuSsOfCFCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T00:32:37.274952Z"},"content_sha256":"e2c57d19abf0fc2dded197d64150e6fb6188b129379bf8b53c6cbc1ac5d3f9fa","schema_version":"1.0","event_id":"sha256:e2c57d19abf0fc2dded197d64150e6fb6188b129379bf8b53c6cbc1ac5d3f9fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EQ6464QR7V26OGER2E27YY6YPZ/bundle.json","state_url":"https://pith.science/pith/EQ6464QR7V26OGER2E27YY6YPZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EQ6464QR7V26OGER2E27YY6YPZ/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":"2026-07-04T00:32:37Z","links":{"resolver":"https://pith.science/pith/EQ6464QR7V26OGER2E27YY6YPZ","bundle":"https://pith.science/pith/EQ6464QR7V26OGER2E27YY6YPZ/bundle.json","state":"https://pith.science/pith/EQ6464QR7V26OGER2E27YY6YPZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EQ6464QR7V26OGER2E27YY6YPZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:EQ6464QR7V26OGER2E27YY6YPZ","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":"83fedc8cbfa1417987fd37d3c1cb089ab30d0ebb20f4d724f038c7f12fe94b99","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-16T04:51:11Z","title_canon_sha256":"a702214f6da8cb3837ad7528ae427626e9173d141f91f35e54d89504bae345c2"},"schema_version":"1.0","source":{"id":"1601.04123","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.04123","created_at":"2026-05-18T00:44:58Z"},{"alias_kind":"arxiv_version","alias_value":"1601.04123v4","created_at":"2026-05-18T00:44:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04123","created_at":"2026-05-18T00:44:58Z"},{"alias_kind":"pith_short_12","alias_value":"EQ6464QR7V26","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"EQ6464QR7V26OGER","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"EQ6464QR","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:e2c57d19abf0fc2dded197d64150e6fb6188b129379bf8b53c6cbc1ac5d3f9fa","target":"graph","created_at":"2026-05-18T00:44:58Z","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 show that a large number of Thom spectra, i.e. colimits of morphisms $BG\\to BGL_1(\\mathbb{S})$, can be obtained as iterated Thom spectra, i.e. colimits of morphisms $BG\\to BGL_1(Mf)$ for some Thom spectrum $Mf$. This leads to a number of new relative Thom isomorphisms, e.g. $MU[6,\\infty)\\wedge_{MString} MU[6,\\infty)\\simeq MU[6,\\infty)\\wedge\\mathbb{S}[B^3Spin]$. As an example of interest to chromatic homotopy theorists, we also show that Ravenel's $X(n)$ filtration of $MU$ is a tower of intermediate Thom spectra determined by a natural filtration of $BU$ by sub-bialagebras.","authors_text":"Jonathan Beardsley","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-16T04:51:11Z","title":"Relative Thom Spectra Via Operadic Kan Extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04123","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":"sha256:7371eddb6c1dab5dbe655576b6717611a357b2e8cb8f39088e2df86b9c6bc8dc","target":"record","created_at":"2026-05-18T00:44:58Z","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":"83fedc8cbfa1417987fd37d3c1cb089ab30d0ebb20f4d724f038c7f12fe94b99","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-16T04:51:11Z","title_canon_sha256":"a702214f6da8cb3837ad7528ae427626e9173d141f91f35e54d89504bae345c2"},"schema_version":"1.0","source":{"id":"1601.04123","kind":"arxiv","version":4}},"canonical_sha256":"243dcf7211fd75e71891d135fc63d87e40570558439cd3702a5bc1dbc68df9b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"243dcf7211fd75e71891d135fc63d87e40570558439cd3702a5bc1dbc68df9b4","first_computed_at":"2026-05-18T00:44:58.195392Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:58.195392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ox0lgDgJDrdHrcuJise1d2bMwLdMgqzxvIm96Ij81xwGNop7snzfT6FUofUa6E4MdsNFuAwzSCmHydJtlfbXAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:58.195836Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.04123","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7371eddb6c1dab5dbe655576b6717611a357b2e8cb8f39088e2df86b9c6bc8dc","sha256:e2c57d19abf0fc2dded197d64150e6fb6188b129379bf8b53c6cbc1ac5d3f9fa"],"state_sha256":"e9477190c3b1faad701ae853a5151815d20d32ac2e9803494629e83048be2696"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+gbQ5m/3CwKyHGRQqL7JgZUh2VA4HhXVCd9Kgm9moaehz6ZMKU489iMXYXi/vsXc6GGtWDM9Rc1j69aHufrEAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T00:32:37.277451Z","bundle_sha256":"c817d8fe7aa097ee459d530298f6c39522dd90bbe03757eb0b23e6eaf2d87eb7"}}