{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:YQ4RPKPCSKXUZ2YFSWIARQTYGE","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":"2ad9e0f04781389a1d9e4c849a078f140854ffd80c5672d00516dc8dd94a2150","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2004-03-25T21:41:37Z","title_canon_sha256":"27eb9c237ce30a2814c65d1f98ba6c7ee9e76e9371ee54a149eb23e87a390c59"},"schema_version":"1.0","source":{"id":"math/0403451","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0403451","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"arxiv_version","alias_value":"math/0403451v2","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0403451","created_at":"2026-05-18T02:41:32Z"},{"alias_kind":"pith_short_12","alias_value":"YQ4RPKPCSKXU","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"YQ4RPKPCSKXUZ2YF","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"YQ4RPKPC","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:262de6a572720b39df6e4c9075b287e745fbf8596e009eb73290e16a7addabdd","target":"graph","created_at":"2026-05-18T02:41:32Z","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":"Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality w","authors_text":"Daniel C. Isaksen, J. Daniel Christensen","cross_cats":[],"headline":"","license":"","primary_cat":"math.AT","submitted_at":"2004-03-25T21:41:37Z","title":"Duality and Pro-Spectra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0403451","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":"sha256:35a561d1602df38041e510e4753975b6293c6241f6437673ff90640f97747d1d","target":"record","created_at":"2026-05-18T02:41:32Z","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":"2ad9e0f04781389a1d9e4c849a078f140854ffd80c5672d00516dc8dd94a2150","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2004-03-25T21:41:37Z","title_canon_sha256":"27eb9c237ce30a2814c65d1f98ba6c7ee9e76e9371ee54a149eb23e87a390c59"},"schema_version":"1.0","source":{"id":"math/0403451","kind":"arxiv","version":2}},"canonical_sha256":"c43917a9e292af4ceb05959008c278310e59b1e9cfedc47b2bea89e7e04d7f84","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c43917a9e292af4ceb05959008c278310e59b1e9cfedc47b2bea89e7e04d7f84","first_computed_at":"2026-05-18T02:41:32.488985Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:32.488985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Js5tqccqVv7qbqY5IEnRb/rIuL8E+FoLJtyuqdGlZor5Ms0GOFjR8DdiEL/VnGQp5Lpaaz7utaKUv+y0nn59BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:32.489433Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0403451","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35a561d1602df38041e510e4753975b6293c6241f6437673ff90640f97747d1d","sha256:262de6a572720b39df6e4c9075b287e745fbf8596e009eb73290e16a7addabdd"],"state_sha256":"fac89ae79e52cd9e5d901e88a594f22fcefb190590d421c7267d79739d3df30f"}