{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:OQD6CHQP3BB6RJ67C7ZC2GQRE4","short_pith_number":"pith:OQD6CHQP","schema_version":"1.0","canonical_sha256":"7407e11e0fd843e8a7df17f22d1a112725b861e78c7fc8636bc67de6b8a0e557","source":{"kind":"arxiv","id":"1711.08579","version":1},"attestation_state":"computed","paper":{"title":"CW-complexes in the Category of Small Categories","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.AT","math.KT"],"primary_cat":"math.CT","authors_text":"Andrew Salch, Christian Frank","submitted_at":"2017-11-23T05:25:45Z","abstract_excerpt":"We compute the collection of CW-complexes in the model category of small categories constructed by Joyal and Tierney. More generally, if $X$ is a connected topological space, we show that the homotopy category of CW-complexes in Joyal-Tierney's model category of sheaves of sets on $X$ is equivalent to the homotopy category of groupoids. As an application of the ideas, we show that the algebraic $K$-theory groups of the category of pointed small categories are trivial, and more generally, the algebraic $K$-theory groups of any sufficiently \"nice\" Waldhausen category $\\mathcal{A}$ of pointed sma"},"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":"1711.08579","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CT","submitted_at":"2017-11-23T05:25:45Z","cross_cats_sorted":["math.AT","math.KT"],"title_canon_sha256":"1800fbd2aa1eb8035deb5b58a3e4adbba8ecabd8584f91ce1c27f43a9e908375","abstract_canon_sha256":"db3b983bf4106cb51d0f695c26675a7e5f1523edc20e1ae803b7078f288c0045"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:44.652819Z","signature_b64":"n1ChbCZgEiok1aK4ZqX/ecThGQSWhXBOErBSN1mnXBGU7ZsxIQQLY3Nk7D9WnlzkTf1B/Ew15uCirA7Dl3vyCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7407e11e0fd843e8a7df17f22d1a112725b861e78c7fc8636bc67de6b8a0e557","last_reissued_at":"2026-05-18T00:29:44.652161Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:44.652161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"CW-complexes in the Category of Small Categories","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.AT","math.KT"],"primary_cat":"math.CT","authors_text":"Andrew Salch, Christian Frank","submitted_at":"2017-11-23T05:25:45Z","abstract_excerpt":"We compute the collection of CW-complexes in the model category of small categories constructed by Joyal and Tierney. More generally, if $X$ is a connected topological space, we show that the homotopy category of CW-complexes in Joyal-Tierney's model category of sheaves of sets on $X$ is equivalent to the homotopy category of groupoids. As an application of the ideas, we show that the algebraic $K$-theory groups of the category of pointed small categories are trivial, and more generally, the algebraic $K$-theory groups of any sufficiently \"nice\" Waldhausen category $\\mathcal{A}$ of pointed sma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08579","kind":"arxiv","version":1},"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":"1711.08579","created_at":"2026-05-18T00:29:44.652251+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.08579v1","created_at":"2026-05-18T00:29:44.652251+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.08579","created_at":"2026-05-18T00:29:44.652251+00:00"},{"alias_kind":"pith_short_12","alias_value":"OQD6CHQP3BB6","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"OQD6CHQP3BB6RJ67","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"OQD6CHQP","created_at":"2026-05-18T12:31:34.259226+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/OQD6CHQP3BB6RJ67C7ZC2GQRE4","json":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4.json","graph_json":"https://pith.science/api/pith-number/OQD6CHQP3BB6RJ67C7ZC2GQRE4/graph.json","events_json":"https://pith.science/api/pith-number/OQD6CHQP3BB6RJ67C7ZC2GQRE4/events.json","paper":"https://pith.science/paper/OQD6CHQP"},"agent_actions":{"view_html":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4","download_json":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4.json","view_paper":"https://pith.science/paper/OQD6CHQP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.08579&json=true","fetch_graph":"https://pith.science/api/pith-number/OQD6CHQP3BB6RJ67C7ZC2GQRE4/graph.json","fetch_events":"https://pith.science/api/pith-number/OQD6CHQP3BB6RJ67C7ZC2GQRE4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4/action/storage_attestation","attest_author":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4/action/author_attestation","sign_citation":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4/action/citation_signature","submit_replication":"https://pith.science/pith/OQD6CHQP3BB6RJ67C7ZC2GQRE4/action/replication_record"}},"created_at":"2026-05-18T00:29:44.652251+00:00","updated_at":"2026-05-18T00:29:44.652251+00:00"}