{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:OPCR27HNDGRRBEXQAFAO3CTFHZ","short_pith_number":"pith:OPCR27HN","schema_version":"1.0","canonical_sha256":"73c51d7ced19a31092f00140ed8a653e7d6bfc7c9ab88a2ca2ecb9f9d56e4c2d","source":{"kind":"arxiv","id":"1306.3915","version":2},"attestation_state":"computed","paper":{"title":"Discrete Homology Theory for Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.MG","authors_text":"Helene Barcelo, Jacob A. White, Valerio Capraro","submitted_at":"2013-06-17T16:17:00Z","abstract_excerpt":"In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that "},"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":"1306.3915","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-06-17T16:17:00Z","cross_cats_sorted":["math.AT","math.CO"],"title_canon_sha256":"8e6ea94b1de9ff63e44b2d4246e582c19a472d1e029b02ab11085af36ae1836d","abstract_canon_sha256":"6a4fe682430b9e1ddd456fe7f5e03120c563a9828c7dd5cc626c1a67f0f503c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:29.724385Z","signature_b64":"2Ta/qyqCIPlju1QWGjPmDMsWGX6AVX5JA6wPaedmTW5Pe7w0QUKn7pUh6fA4kwutfieQ0/lc5tQrKb+i+fUiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73c51d7ced19a31092f00140ed8a653e7d6bfc7c9ab88a2ca2ecb9f9d56e4c2d","last_reissued_at":"2026-05-18T00:44:29.723865Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:29.723865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrete Homology Theory for Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.MG","authors_text":"Helene Barcelo, Jacob A. White, Valerio Capraro","submitted_at":"2013-06-17T16:17:00Z","abstract_excerpt":"In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3915","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1306.3915","created_at":"2026-05-18T00:44:29.723937+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.3915v2","created_at":"2026-05-18T00:44:29.723937+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.3915","created_at":"2026-05-18T00:44:29.723937+00:00"},{"alias_kind":"pith_short_12","alias_value":"OPCR27HNDGRR","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"OPCR27HNDGRRBEXQ","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"OPCR27HN","created_at":"2026-05-18T12:27:54.935989+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/OPCR27HNDGRRBEXQAFAO3CTFHZ","json":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ.json","graph_json":"https://pith.science/api/pith-number/OPCR27HNDGRRBEXQAFAO3CTFHZ/graph.json","events_json":"https://pith.science/api/pith-number/OPCR27HNDGRRBEXQAFAO3CTFHZ/events.json","paper":"https://pith.science/paper/OPCR27HN"},"agent_actions":{"view_html":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ","download_json":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ.json","view_paper":"https://pith.science/paper/OPCR27HN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.3915&json=true","fetch_graph":"https://pith.science/api/pith-number/OPCR27HNDGRRBEXQAFAO3CTFHZ/graph.json","fetch_events":"https://pith.science/api/pith-number/OPCR27HNDGRRBEXQAFAO3CTFHZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ/action/storage_attestation","attest_author":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ/action/author_attestation","sign_citation":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ/action/citation_signature","submit_replication":"https://pith.science/pith/OPCR27HNDGRRBEXQAFAO3CTFHZ/action/replication_record"}},"created_at":"2026-05-18T00:44:29.723937+00:00","updated_at":"2026-05-18T00:44:29.723937+00:00"}