{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:IXBJVVELGLNHAQCINK6AH6SJMY","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":"cf17989aa334fb86e41193d3b327837112a6979dbceaffed2b3c7c2c4cae092e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-09-11T18:01:09Z","title_canon_sha256":"2d72969e416f627f69e75d97719a3b8b60fb4e6b1b02e5759a2cb61d2277316e"},"schema_version":"1.0","source":{"id":"1209.2384","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.2384","created_at":"2026-05-18T01:25:55Z"},{"alias_kind":"arxiv_version","alias_value":"1209.2384v4","created_at":"2026-05-18T01:25:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2384","created_at":"2026-05-18T01:25:55Z"},{"alias_kind":"pith_short_12","alias_value":"IXBJVVELGLNH","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"IXBJVVELGLNHAQCI","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"IXBJVVEL","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:cec539d26ef2c529f97a10e8e7db651d6ec108b695f1c023483383d31769c22f","target":"graph","created_at":"2026-05-18T01:25:55Z","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":"In this paper, we show that for reduced homotopy endofunctors of spaces, F, and for all $n \\geq 1$ there are adjoint functors $R_n, L_n$ with $T_n F \\simeq R_n F L_n$, where $P_n F$ is the $n$-excisive approximation to $F$, constructed by taking the homotopy colimit over iterations of $T_n F$. This then endows $T_n$ of the identity with the structure of a monad and the $T_n F$'s are the functor version of bimodules over that monad. It follows that each $T_n F$ (and $P_nF$) takes values in spaces of symmetric Lusternik-Schnirelman cocategory $n$, as defined by Hopkins. This also recovers recent","authors_text":"Rosona Eldred","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-09-11T18:01:09Z","title":"Goodwillie Calculus via Adjunction and LS Cocategory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2384","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:d4ec4594b3c03b19e3f4c86b527ff1907ae465947750eed28202bdbe7ddc57f4","target":"record","created_at":"2026-05-18T01:25:55Z","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":"cf17989aa334fb86e41193d3b327837112a6979dbceaffed2b3c7c2c4cae092e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-09-11T18:01:09Z","title_canon_sha256":"2d72969e416f627f69e75d97719a3b8b60fb4e6b1b02e5759a2cb61d2277316e"},"schema_version":"1.0","source":{"id":"1209.2384","kind":"arxiv","version":4}},"canonical_sha256":"45c29ad48b32da7040486abc03fa496600dc0533b8b4593864949a36382cb57c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45c29ad48b32da7040486abc03fa496600dc0533b8b4593864949a36382cb57c","first_computed_at":"2026-05-18T01:25:55.975855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:55.975855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XUu/5xqM8+aOMZCcdG8Qo5Gz4kCQXfqT8dWqbwTqNOQ350XodxMGrVsvuQQjaohwvXTUL8Z47E0tunNkEBxrBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:55.976531Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.2384","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4ec4594b3c03b19e3f4c86b527ff1907ae465947750eed28202bdbe7ddc57f4","sha256:cec539d26ef2c529f97a10e8e7db651d6ec108b695f1c023483383d31769c22f"],"state_sha256":"a3ed9d6f6934be11819af0e81427e0d00fc34972b1231979137711731462b212"}