{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:ZGLQELEKPW7LOHV4WNRD427KCR","short_pith_number":"pith:ZGLQELEK","canonical_record":{"source":{"id":"2402.12339","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-02-19T18:11:58Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"4ad165bf4939cc20968db2091969291de7675c8bf07f18ffac2c74e3b9e3b084","abstract_canon_sha256":"c1400fb9255c3214fdcf12cac19b8b5c6fa54a2396de32e38c020627853d855b"},"schema_version":"1.0"},"canonical_sha256":"c997022c8a7dbeb71ebcb3623e6bea145926a02182792591cec82d9f75160126","source":{"kind":"arxiv","id":"2402.12339","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.12339","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"arxiv_version","alias_value":"2402.12339v2","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.12339","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"pith_short_12","alias_value":"ZGLQELEKPW7L","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"pith_short_16","alias_value":"ZGLQELEKPW7LOHV4","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"pith_short_8","alias_value":"ZGLQELEK","created_at":"2026-07-05T11:48:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:ZGLQELEKPW7LOHV4WNRD427KCR","target":"record","payload":{"canonical_record":{"source":{"id":"2402.12339","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-02-19T18:11:58Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"4ad165bf4939cc20968db2091969291de7675c8bf07f18ffac2c74e3b9e3b084","abstract_canon_sha256":"c1400fb9255c3214fdcf12cac19b8b5c6fa54a2396de32e38c020627853d855b"},"schema_version":"1.0"},"canonical_sha256":"c997022c8a7dbeb71ebcb3623e6bea145926a02182792591cec82d9f75160126","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:48:45.356199Z","signature_b64":"z9ZBAy4bnYG0zyLOuAVL2mofgp1/GXRy9tFwYHQcw2FKYaqF1veICL2ETmtUX6b8je1mbVFTauaVpGxwSdDhDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c997022c8a7dbeb71ebcb3623e6bea145926a02182792591cec82d9f75160126","last_reissued_at":"2026-07-05T11:48:45.355626Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:48:45.355626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.12339","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":"2026-07-05T11:48:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q3TFfW9WgF2g2yaafYw7+zNdacScE/+gshX79y2gqMyABBcJCC0ASJifRj8lX50w/MoVY+Wd7bP3K6k89k+kAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T11:48:36.009837Z"},"content_sha256":"de12774d294e036898fcc28230da47024722a03e544a5ae943169315cdf1e19e","schema_version":"1.0","event_id":"sha256:de12774d294e036898fcc28230da47024722a03e544a5ae943169315cdf1e19e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:ZGLQELEKPW7LOHV4WNRD427KCR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Path spaces of pushouts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"David W\\\"arn","submitted_at":"2024-02-19T18:11:58Z","abstract_excerpt":"Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the zigzag construction. We also obtain a description of loop spaces of homotopy pushouts. Using the zigzag construction, we reproduce generalisations of the Blakers-Massey theorem and fundamental results from Bass-Serre theory. We also describe the loop space of a wedge and show that it splits after suspension. Our construction can be interpreted in a large class"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.12339","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2402.12339/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":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:48:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kj6Xh45qWNO9lgfmHXjJw/Z6lEBOY5b4YEK9l/MSkwQKLUFfa6TxdXApl9dmRfWFqYObSDi8g3SCywgk62DSCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T11:48:36.011168Z"},"content_sha256":"564c4417ec19009cdeefb6d3a9a0b2d14d2dc291cfaa1c0e25a76aa2142b8281","schema_version":"1.0","event_id":"sha256:564c4417ec19009cdeefb6d3a9a0b2d14d2dc291cfaa1c0e25a76aa2142b8281"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZGLQELEKPW7LOHV4WNRD427KCR/bundle.json","state_url":"https://pith.science/pith/ZGLQELEKPW7LOHV4WNRD427KCR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZGLQELEKPW7LOHV4WNRD427KCR/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":"2026-08-11T11:48:36Z","links":{"resolver":"https://pith.science/pith/ZGLQELEKPW7LOHV4WNRD427KCR","bundle":"https://pith.science/pith/ZGLQELEKPW7LOHV4WNRD427KCR/bundle.json","state":"https://pith.science/pith/ZGLQELEKPW7LOHV4WNRD427KCR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZGLQELEKPW7LOHV4WNRD427KCR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:ZGLQELEKPW7LOHV4WNRD427KCR","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":"c1400fb9255c3214fdcf12cac19b8b5c6fa54a2396de32e38c020627853d855b","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-02-19T18:11:58Z","title_canon_sha256":"4ad165bf4939cc20968db2091969291de7675c8bf07f18ffac2c74e3b9e3b084"},"schema_version":"1.0","source":{"id":"2402.12339","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.12339","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"arxiv_version","alias_value":"2402.12339v2","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.12339","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"pith_short_12","alias_value":"ZGLQELEKPW7L","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"pith_short_16","alias_value":"ZGLQELEKPW7LOHV4","created_at":"2026-07-05T11:48:45Z"},{"alias_kind":"pith_short_8","alias_value":"ZGLQELEK","created_at":"2026-07-05T11:48:45Z"}],"graph_snapshots":[{"event_id":"sha256:564c4417ec19009cdeefb6d3a9a0b2d14d2dc291cfaa1c0e25a76aa2142b8281","target":"graph","created_at":"2026-07-05T11:48:45Z","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/2402.12339/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the zigzag construction. We also obtain a description of loop spaces of homotopy pushouts. Using the zigzag construction, we reproduce generalisations of the Blakers-Massey theorem and fundamental results from Bass-Serre theory. We also describe the loop space of a wedge and show that it splits after suspension. Our construction can be interpreted in a large class","authors_text":"David W\\\"arn","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-02-19T18:11:58Z","title":"Path spaces of pushouts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.12339","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:de12774d294e036898fcc28230da47024722a03e544a5ae943169315cdf1e19e","target":"record","created_at":"2026-07-05T11:48:45Z","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":"c1400fb9255c3214fdcf12cac19b8b5c6fa54a2396de32e38c020627853d855b","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-02-19T18:11:58Z","title_canon_sha256":"4ad165bf4939cc20968db2091969291de7675c8bf07f18ffac2c74e3b9e3b084"},"schema_version":"1.0","source":{"id":"2402.12339","kind":"arxiv","version":2}},"canonical_sha256":"c997022c8a7dbeb71ebcb3623e6bea145926a02182792591cec82d9f75160126","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c997022c8a7dbeb71ebcb3623e6bea145926a02182792591cec82d9f75160126","first_computed_at":"2026-07-05T11:48:45.355626Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:48:45.355626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z9ZBAy4bnYG0zyLOuAVL2mofgp1/GXRy9tFwYHQcw2FKYaqF1veICL2ETmtUX6b8je1mbVFTauaVpGxwSdDhDg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:48:45.356199Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.12339","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de12774d294e036898fcc28230da47024722a03e544a5ae943169315cdf1e19e","sha256:564c4417ec19009cdeefb6d3a9a0b2d14d2dc291cfaa1c0e25a76aa2142b8281"],"state_sha256":"95d922cfbfa1a579cf11b110900ee6c22b69123d473204c9eece368da20a879d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4IKMSY4EKpYWT3ul1exNKftbAdm3vYFm2pekCtOSjHNxrFwvOEgGdwiwXdlxK2uYTO9+ScBC98a7fAwx0MpQAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T11:48:36.018857Z","bundle_sha256":"3cf2ad95a473e430972824959169feb532edc6c7ba89b79dfc47fbc7a11618dd"}}