{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:JQ2BF73YHCMNCL4O6W46R6LBFQ","short_pith_number":"pith:JQ2BF73Y","canonical_record":{"source":{"id":"1612.07762","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-22T19:28:44Z","cross_cats_sorted":[],"title_canon_sha256":"5eb7ec7f55513bcdfae16d6c64a1ff5e0c7ec07ff5c03d172c10953edeeec29d","abstract_canon_sha256":"845d5c1eab4a86867171b0c869022b6a106946f45753fbde8b3f9af12269d9da"},"schema_version":"1.0"},"canonical_sha256":"4c3412ff783898d12f8ef5b9e8f9612c18ecc192f685a5f7774d6017a2ca1fd1","source":{"kind":"arxiv","id":"1612.07762","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.07762","created_at":"2026-05-17T23:58:42Z"},{"alias_kind":"arxiv_version","alias_value":"1612.07762v2","created_at":"2026-05-17T23:58:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.07762","created_at":"2026-05-17T23:58:42Z"},{"alias_kind":"pith_short_12","alias_value":"JQ2BF73YHCMN","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JQ2BF73YHCMNCL4O","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JQ2BF73Y","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:JQ2BF73YHCMNCL4O6W46R6LBFQ","target":"record","payload":{"canonical_record":{"source":{"id":"1612.07762","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-22T19:28:44Z","cross_cats_sorted":[],"title_canon_sha256":"5eb7ec7f55513bcdfae16d6c64a1ff5e0c7ec07ff5c03d172c10953edeeec29d","abstract_canon_sha256":"845d5c1eab4a86867171b0c869022b6a106946f45753fbde8b3f9af12269d9da"},"schema_version":"1.0"},"canonical_sha256":"4c3412ff783898d12f8ef5b9e8f9612c18ecc192f685a5f7774d6017a2ca1fd1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:42.580269Z","signature_b64":"fcKldm7jAWrvHeNL1bswsd2p/JmHfESIGJ07LepwdxLP8+/CqihZKg1lggpEv+j+VFQNsG45RO/1naHW0vVJAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c3412ff783898d12f8ef5b9e8f9612c18ecc192f685a5f7774d6017a2ca1fd1","last_reissued_at":"2026-05-17T23:58:42.579822Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:42.579822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.07762","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":"2026-05-17T23:58:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nsPwtaIedbDMRb+VvwUaiBU/+SR3lt8YxQQTljFcM1Jyq3wSShtCAxyfadGUehWyob18745FdISpESPQ6onRBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T17:50:30.015803Z"},"content_sha256":"fd58a476340aaa9e8492f09f9fd95865670b470d7d93deb0a95ade980f9cf7fd","schema_version":"1.0","event_id":"sha256:fd58a476340aaa9e8492f09f9fd95865670b470d7d93deb0a95ade980f9cf7fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:JQ2BF73YHCMNCL4O6W46R6LBFQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic Hopf invariants and rational models for mapping spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Felix Wierstra","submitted_at":"2016-12-22T19:28:44Z","abstract_excerpt":"In this paper we will define an invariant $mc_{\\infty}(f)$ of maps $f:X \\rightarrow Y_{\\mathbb{Q}}$ between a finite CW-complex and a rational space $Y_{\\mathbb{Q}}$. We prove that this invariant is complete, i.e. $mc_{\\infty}(f)=mc_{\\infty}(g)$ if an only if $f$ and $g$ are homotopic. We will also construct an $L_{\\infty}$-model for the based mapping space $Map_*(X,Y_{\\mathbb{Q}})$ from a $C_{\\infty}$-coalgebra and an $L_{\\infty}$-algebra."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07762","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":"2026-05-17T23:58:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9sndSK3YbKJmgRQ+K9TOQMlieYUTD1xGTOxyVxolq6390+6jW4AySF7du28+A4zMz+KaDTnpRLbf/JudhulJDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T17:50:30.016149Z"},"content_sha256":"531eeffe86163dfe741657d9dada8b817c76e9ae98d74f361e159d31f83d8b90","schema_version":"1.0","event_id":"sha256:531eeffe86163dfe741657d9dada8b817c76e9ae98d74f361e159d31f83d8b90"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JQ2BF73YHCMNCL4O6W46R6LBFQ/bundle.json","state_url":"https://pith.science/pith/JQ2BF73YHCMNCL4O6W46R6LBFQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JQ2BF73YHCMNCL4O6W46R6LBFQ/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":"2026-06-24T17:50:30Z","links":{"resolver":"https://pith.science/pith/JQ2BF73YHCMNCL4O6W46R6LBFQ","bundle":"https://pith.science/pith/JQ2BF73YHCMNCL4O6W46R6LBFQ/bundle.json","state":"https://pith.science/pith/JQ2BF73YHCMNCL4O6W46R6LBFQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JQ2BF73YHCMNCL4O6W46R6LBFQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:JQ2BF73YHCMNCL4O6W46R6LBFQ","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":"845d5c1eab4a86867171b0c869022b6a106946f45753fbde8b3f9af12269d9da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-22T19:28:44Z","title_canon_sha256":"5eb7ec7f55513bcdfae16d6c64a1ff5e0c7ec07ff5c03d172c10953edeeec29d"},"schema_version":"1.0","source":{"id":"1612.07762","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.07762","created_at":"2026-05-17T23:58:42Z"},{"alias_kind":"arxiv_version","alias_value":"1612.07762v2","created_at":"2026-05-17T23:58:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.07762","created_at":"2026-05-17T23:58:42Z"},{"alias_kind":"pith_short_12","alias_value":"JQ2BF73YHCMN","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JQ2BF73YHCMNCL4O","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JQ2BF73Y","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:531eeffe86163dfe741657d9dada8b817c76e9ae98d74f361e159d31f83d8b90","target":"graph","created_at":"2026-05-17T23:58:42Z","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 will define an invariant $mc_{\\infty}(f)$ of maps $f:X \\rightarrow Y_{\\mathbb{Q}}$ between a finite CW-complex and a rational space $Y_{\\mathbb{Q}}$. We prove that this invariant is complete, i.e. $mc_{\\infty}(f)=mc_{\\infty}(g)$ if an only if $f$ and $g$ are homotopic. We will also construct an $L_{\\infty}$-model for the based mapping space $Map_*(X,Y_{\\mathbb{Q}})$ from a $C_{\\infty}$-coalgebra and an $L_{\\infty}$-algebra.","authors_text":"Felix Wierstra","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-22T19:28:44Z","title":"Algebraic Hopf invariants and rational models for mapping spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07762","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:fd58a476340aaa9e8492f09f9fd95865670b470d7d93deb0a95ade980f9cf7fd","target":"record","created_at":"2026-05-17T23:58:42Z","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":"845d5c1eab4a86867171b0c869022b6a106946f45753fbde8b3f9af12269d9da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-22T19:28:44Z","title_canon_sha256":"5eb7ec7f55513bcdfae16d6c64a1ff5e0c7ec07ff5c03d172c10953edeeec29d"},"schema_version":"1.0","source":{"id":"1612.07762","kind":"arxiv","version":2}},"canonical_sha256":"4c3412ff783898d12f8ef5b9e8f9612c18ecc192f685a5f7774d6017a2ca1fd1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c3412ff783898d12f8ef5b9e8f9612c18ecc192f685a5f7774d6017a2ca1fd1","first_computed_at":"2026-05-17T23:58:42.579822Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:42.579822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fcKldm7jAWrvHeNL1bswsd2p/JmHfESIGJ07LepwdxLP8+/CqihZKg1lggpEv+j+VFQNsG45RO/1naHW0vVJAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:42.580269Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.07762","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fd58a476340aaa9e8492f09f9fd95865670b470d7d93deb0a95ade980f9cf7fd","sha256:531eeffe86163dfe741657d9dada8b817c76e9ae98d74f361e159d31f83d8b90"],"state_sha256":"69e8d432d3abcceb4e10edfe2e6dc047e8b7e51e1f84e1e160bfdd8e7c8cc099"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"35KdvYRbi11iwsAS2YOhACNz6Jx4pm8RTuf7K5zEB8/lANu9OO7NG0piHXZ/cuCNbIToWJL9/pYr6zlwa1/aAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T17:50:30.018163Z","bundle_sha256":"5043717cb3d73ea959424a58c5cb9667fd1ed9b722d34e5fd36e48f88b214fa7"}}