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The family of constraints of $A$ induces a {\\it structure} on $(R,M).$ The main result of the paper is:\n  1. {\\it There exists a bijection between the set of structures on $(R,M)$ and the group of Mac Lane 3-cocycles $Z^{3}_{MaL}(R, M).$}\n  2. {\\it There exists a bijection between $C(R,M)$ of congruence classes of Ann-categories whose pre-stick is of the type $(R,M)$ and the Mac Lane cohomology group $H^3_{\\textrm{MaL}}(R,M).$}"},"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":"0805.1505","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2008-05-11T02:51:02Z","cross_cats_sorted":[],"title_canon_sha256":"b95e483068922062158fa0153e1418b35b04669cf8de6ebf7873a80ae441f757","abstract_canon_sha256":"42eca3e40bc7561c1841298cbf83f4e721079ae5f0800c8b47c3aa15751ff192"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:22.230739Z","signature_b64":"FfEZZS2s/arW2PcjpH128MM0z8F0a+YITHRTTyf1mrqCtV+BHF2a5BpCzj6tc68mc4Xfe8gGpuUgiBd2P576DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bd7501115b7cc8c6647c3f658c52d16e98d18852034934ef4235d076077b3de","last_reissued_at":"2026-05-18T03:13:22.229747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:22.229747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structure of Ann-categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Nguyen Tien Quang","submitted_at":"2008-05-11T02:51:02Z","abstract_excerpt":"Each Ann-category $\\A$ is equivalent to an Ann-category of the type $(R,M),$ where $M$ is an $R$-bimodule. 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