{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:4SPRFNCRVE6SRKPUPWDFFXXQE5","short_pith_number":"pith:4SPRFNCR","schema_version":"1.0","canonical_sha256":"e49f12b451a93d28a9f47d8652def027498ada2b7f26ecc6294b6c30619d6ebb","source":{"kind":"arxiv","id":"0907.4553","version":1},"attestation_state":"computed","paper":{"title":"Coherence for weak units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Andr\\'e Joyal, Joachim Kock","submitted_at":"2009-07-27T06:59:09Z","abstract_excerpt":"We define weak units in a semi-monoidal 2-category $\\CC$ as cancellable pseudo-idempotents: they are pairs $(I,\\alpha)$ where $I$ is an object such that tensoring with $I$ from either side constitutes a biequivalence of $\\CC$, and $\\alpha: I \\tensor I \\to I$ is an equivalence in $\\CC$. We show that this notion of weak unit has coherence built in: Theorem A: $\\alpha$ has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contract"},"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":"0907.4553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2009-07-27T06:59:09Z","cross_cats_sorted":[],"title_canon_sha256":"cbc73b1976466e365c54a1e5ee88d05f7701a14dd2f24b3d59678b3b4429375d","abstract_canon_sha256":"22da63819b399b58d0dd7abe2fcd328914a59013ae8aa0b7046e7c123ffb105e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:51.478644Z","signature_b64":"w5n/OE4XjVMDvNnGN2AzSFd2tfdkJemmoYjt9Pg8M0skL3xVvpohxSbBMDaHj+3XDFMDEHzB08VblQa+2IguAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e49f12b451a93d28a9f47d8652def027498ada2b7f26ecc6294b6c30619d6ebb","last_reissued_at":"2026-05-18T02:47:51.478017Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:51.478017Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coherence for weak units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Andr\\'e Joyal, Joachim Kock","submitted_at":"2009-07-27T06:59:09Z","abstract_excerpt":"We define weak units in a semi-monoidal 2-category $\\CC$ as cancellable pseudo-idempotents: they are pairs $(I,\\alpha)$ where $I$ is an object such that tensoring with $I$ from either side constitutes a biequivalence of $\\CC$, and $\\alpha: I \\tensor I \\to I$ is an equivalence in $\\CC$. We show that this notion of weak unit has coherence built in: Theorem A: $\\alpha$ has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contract"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.4553","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":"0907.4553","created_at":"2026-05-18T02:47:51.478113+00:00"},{"alias_kind":"arxiv_version","alias_value":"0907.4553v1","created_at":"2026-05-18T02:47:51.478113+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.4553","created_at":"2026-05-18T02:47:51.478113+00:00"},{"alias_kind":"pith_short_12","alias_value":"4SPRFNCRVE6S","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"4SPRFNCRVE6SRKPU","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"4SPRFNCR","created_at":"2026-05-18T12:25:58.837520+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/4SPRFNCRVE6SRKPUPWDFFXXQE5","json":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5.json","graph_json":"https://pith.science/api/pith-number/4SPRFNCRVE6SRKPUPWDFFXXQE5/graph.json","events_json":"https://pith.science/api/pith-number/4SPRFNCRVE6SRKPUPWDFFXXQE5/events.json","paper":"https://pith.science/paper/4SPRFNCR"},"agent_actions":{"view_html":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5","download_json":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5.json","view_paper":"https://pith.science/paper/4SPRFNCR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0907.4553&json=true","fetch_graph":"https://pith.science/api/pith-number/4SPRFNCRVE6SRKPUPWDFFXXQE5/graph.json","fetch_events":"https://pith.science/api/pith-number/4SPRFNCRVE6SRKPUPWDFFXXQE5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5/action/storage_attestation","attest_author":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5/action/author_attestation","sign_citation":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5/action/citation_signature","submit_replication":"https://pith.science/pith/4SPRFNCRVE6SRKPUPWDFFXXQE5/action/replication_record"}},"created_at":"2026-05-18T02:47:51.478113+00:00","updated_at":"2026-05-18T02:47:51.478113+00:00"}