{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:GRN6S5C4VFR4TPH5RUC32WQCPJ","short_pith_number":"pith:GRN6S5C4","canonical_record":{"source":{"id":"math/0502308","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2005-02-15T12:44:57Z","cross_cats_sorted":["math.CT","math.RT"],"title_canon_sha256":"c84d75d81bed536d155e487a6858d383da55bdc8a8b1977f551a5ce752b6b361","abstract_canon_sha256":"8015e775f5076c2cf3d2df9ed0b464171de4d851853a725a955deae446027f29"},"schema_version":"1.0"},"canonical_sha256":"345be9745ca963c9bcfd8d05bd5a027a6fa462963c0e82d720133fd27c50d60e","source":{"kind":"arxiv","id":"math/0502308","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0502308","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/0502308v1","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0502308","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"pith_short_12","alias_value":"GRN6S5C4VFR4","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"GRN6S5C4VFR4TPH5","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"GRN6S5C4","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:GRN6S5C4VFR4TPH5RUC32WQCPJ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0502308","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2005-02-15T12:44:57Z","cross_cats_sorted":["math.CT","math.RT"],"title_canon_sha256":"c84d75d81bed536d155e487a6858d383da55bdc8a8b1977f551a5ce752b6b361","abstract_canon_sha256":"8015e775f5076c2cf3d2df9ed0b464171de4d851853a725a955deae446027f29"},"schema_version":"1.0"},"canonical_sha256":"345be9745ca963c9bcfd8d05bd5a027a6fa462963c0e82d720133fd27c50d60e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:51.306097Z","signature_b64":"V2F6+K28/oeATCOfkqenzaxytWwkU/tcAYrIT8vmmgy72DVxUjyPzwMlZcFDQm7Crgr2sQTni1q++ZbMCsSEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"345be9745ca963c9bcfd8d05bd5a027a6fa462963c0e82d720133fd27c50d60e","last_reissued_at":"2026-05-18T01:08:51.305255Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:51.305255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0502308","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kq9tzhxin07/qyDqG4XeG3xrFFltE6D6sI/JynhboUhoFk5PrH6k7EPC0brYI6o9POc8lZYCVC8K1/3P0mGIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T01:36:11.212536Z"},"content_sha256":"1df2c58f080429800c46c62c983d46d34448c2ae887f916fbd2efbace251a1a9","schema_version":"1.0","event_id":"sha256:1df2c58f080429800c46c62c983d46d34448c2ae887f916fbd2efbace251a1a9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:GRN6S5C4VFR4TPH5RUC32WQCPJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Galois coverings, Morita equivalence and smash extensions of categories over a field","license":"","headline":"","cross_cats":["math.CT","math.RT"],"primary_cat":"math.RA","authors_text":"Andrea Solotar, Claude Cibils (Institut de Math\\'ematiques et de Mod\\'elisation de Montpellier)","submitted_at":"2005-02-15T12:44:57Z","abstract_excerpt":"We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502308","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AO56EHJn9wvvOFaCtJpXg5C2JG9yMNIZYFt+vXjXFILX1e6aIgCbrW71ly68cdr9d2GKi6mYaNeSeb375YmTCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T01:36:11.212877Z"},"content_sha256":"b9dbc2350ff531058ea3fecd0daeefe719e548b5cd62169c31051ea189e886b9","schema_version":"1.0","event_id":"sha256:b9dbc2350ff531058ea3fecd0daeefe719e548b5cd62169c31051ea189e886b9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GRN6S5C4VFR4TPH5RUC32WQCPJ/bundle.json","state_url":"https://pith.science/pith/GRN6S5C4VFR4TPH5RUC32WQCPJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GRN6S5C4VFR4TPH5RUC32WQCPJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-29T01:36:11Z","links":{"resolver":"https://pith.science/pith/GRN6S5C4VFR4TPH5RUC32WQCPJ","bundle":"https://pith.science/pith/GRN6S5C4VFR4TPH5RUC32WQCPJ/bundle.json","state":"https://pith.science/pith/GRN6S5C4VFR4TPH5RUC32WQCPJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GRN6S5C4VFR4TPH5RUC32WQCPJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:GRN6S5C4VFR4TPH5RUC32WQCPJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8015e775f5076c2cf3d2df9ed0b464171de4d851853a725a955deae446027f29","cross_cats_sorted":["math.CT","math.RT"],"license":"","primary_cat":"math.RA","submitted_at":"2005-02-15T12:44:57Z","title_canon_sha256":"c84d75d81bed536d155e487a6858d383da55bdc8a8b1977f551a5ce752b6b361"},"schema_version":"1.0","source":{"id":"math/0502308","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0502308","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/0502308v1","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0502308","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"pith_short_12","alias_value":"GRN6S5C4VFR4","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"GRN6S5C4VFR4TPH5","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"GRN6S5C4","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:b9dbc2350ff531058ea3fecd0daeefe719e548b5cd62169c31051ea189e886b9","target":"graph","created_at":"2026-05-18T01:08:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category.","authors_text":"Andrea Solotar, Claude Cibils (Institut de Math\\'ematiques et de Mod\\'elisation de Montpellier)","cross_cats":["math.CT","math.RT"],"headline":"","license":"","primary_cat":"math.RA","submitted_at":"2005-02-15T12:44:57Z","title":"Galois coverings, Morita equivalence and smash extensions of categories over a field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502308","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1df2c58f080429800c46c62c983d46d34448c2ae887f916fbd2efbace251a1a9","target":"record","created_at":"2026-05-18T01:08:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8015e775f5076c2cf3d2df9ed0b464171de4d851853a725a955deae446027f29","cross_cats_sorted":["math.CT","math.RT"],"license":"","primary_cat":"math.RA","submitted_at":"2005-02-15T12:44:57Z","title_canon_sha256":"c84d75d81bed536d155e487a6858d383da55bdc8a8b1977f551a5ce752b6b361"},"schema_version":"1.0","source":{"id":"math/0502308","kind":"arxiv","version":1}},"canonical_sha256":"345be9745ca963c9bcfd8d05bd5a027a6fa462963c0e82d720133fd27c50d60e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"345be9745ca963c9bcfd8d05bd5a027a6fa462963c0e82d720133fd27c50d60e","first_computed_at":"2026-05-18T01:08:51.305255Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:51.305255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V2F6+K28/oeATCOfkqenzaxytWwkU/tcAYrIT8vmmgy72DVxUjyPzwMlZcFDQm7Crgr2sQTni1q++ZbMCsSEBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:51.306097Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0502308","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1df2c58f080429800c46c62c983d46d34448c2ae887f916fbd2efbace251a1a9","sha256:b9dbc2350ff531058ea3fecd0daeefe719e548b5cd62169c31051ea189e886b9"],"state_sha256":"198b4b9b9a015e82a3be3666a58673ed9440ce69ca1a91bd7abae2a4a2cd1fcd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+zmUTtWPb6mETVxsgDoIR5r3P8fUzn8YUx1jT9V6Rf8zrQw50CdF9qGiY1NQgekVwjY1f4pGxrbQ8sbzIyaxAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T01:36:11.214865Z","bundle_sha256":"0c6b58f8176828aa5d4a2f8a5f776560c2cd22f7f5f5a8d1803f2e903e916e73"}}