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We prove that a continuous map of path connected pointed spaces $f: X\\to Y$ is a weak homotopy equivalence if and only if $\\mathbf{C}_*(f): \\mathbf{C}_*(X)\\to \\mathbf{C}_*(Y)$ is an $\\mathbf{\\Omega}$-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebras after applying "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.10237","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-08-30T11:43:04Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"e68ea5ccba5293602fe952a9f8f81819a3276a85094c93e0bae24084aa350a5b","abstract_canon_sha256":"dc254569f982d659359c02f50a50fc02fa770e015fcb3b0e29a280521dff0e52"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:42.601946Z","signature_b64":"Dbh9Hb8/KslZufiW6FvRKRzr5rWY4TOzqMoRDfEFFjnFHvyvtke6LJswcjqUIF3jAkSwbXfB1oX2oVZpn+ddDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46b1ce1d3f3d841b26934e4f2a625f1479ed1c532d281fa92c2fddea301b995e","last_reissued_at":"2026-05-17T23:55:42.601462Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:42.601462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The functor of singular chains detects weak homotopy equivalences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.AT","authors_text":"Felix Wierstra, Mahmoud Zeinalian, Manuel Rivera","submitted_at":"2018-08-30T11:43:04Z","abstract_excerpt":"The normalized singular chains of a path connected pointed space $X$ may be considered as a connected $E_{\\infty}$-coalgebra $\\mathbf{C}_*(X)$ with the property that the $0^{\\text{th}}$ homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode, i.e. it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces $f: X\\to Y$ is a weak homotopy equivalence if and only if $\\mathbf{C}_*(f): \\mathbf{C}_*(X)\\to \\mathbf{C}_*(Y)$ is an $\\mathbf{\\Omega}$-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebras after applying "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10237","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.10237","created_at":"2026-05-17T23:55:42.601542+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.10237v3","created_at":"2026-05-17T23:55:42.601542+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.10237","created_at":"2026-05-17T23:55:42.601542+00:00"},{"alias_kind":"pith_short_12","alias_value":"I2Y44HJ7HWCB","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"I2Y44HJ7HWCBWJUT","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"I2Y44HJ7","created_at":"2026-05-18T12:32:28.185984+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR","json":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR.json","graph_json":"https://pith.science/api/pith-number/I2Y44HJ7HWCBWJUTJZHSUYS7CR/graph.json","events_json":"https://pith.science/api/pith-number/I2Y44HJ7HWCBWJUTJZHSUYS7CR/events.json","paper":"https://pith.science/paper/I2Y44HJ7"},"agent_actions":{"view_html":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR","download_json":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR.json","view_paper":"https://pith.science/paper/I2Y44HJ7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.10237&json=true","fetch_graph":"https://pith.science/api/pith-number/I2Y44HJ7HWCBWJUTJZHSUYS7CR/graph.json","fetch_events":"https://pith.science/api/pith-number/I2Y44HJ7HWCBWJUTJZHSUYS7CR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR/action/storage_attestation","attest_author":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR/action/author_attestation","sign_citation":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR/action/citation_signature","submit_replication":"https://pith.science/pith/I2Y44HJ7HWCBWJUTJZHSUYS7CR/action/replication_record"}},"created_at":"2026-05-17T23:55:42.601542+00:00","updated_at":"2026-05-17T23:55:42.601542+00:00"}