{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:BEO4AMLSJ5OCOA26FXIQMTLBPV","short_pith_number":"pith:BEO4AMLS","canonical_record":{"source":{"id":"1704.07257","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2017-04-24T14:39:07Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"3404332046f8abbe4036086894531226f3f7da3b76db61428d978ddfc6cb133a","abstract_canon_sha256":"1555e2b4375a9332917babd32ea066f7b3c563f8c89255ae196a20ea3b0ebcf1"},"schema_version":"1.0"},"canonical_sha256":"091dc031724f5c27035e2dd1064d617d42ba623472a0214ab98df4f165e959a0","source":{"kind":"arxiv","id":"1704.07257","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.07257","created_at":"2026-05-18T00:20:27Z"},{"alias_kind":"arxiv_version","alias_value":"1704.07257v1","created_at":"2026-05-18T00:20:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.07257","created_at":"2026-05-18T00:20:27Z"},{"alias_kind":"pith_short_12","alias_value":"BEO4AMLSJ5OC","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BEO4AMLSJ5OCOA26","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BEO4AMLS","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:BEO4AMLSJ5OCOA26FXIQMTLBPV","target":"record","payload":{"canonical_record":{"source":{"id":"1704.07257","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2017-04-24T14:39:07Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"3404332046f8abbe4036086894531226f3f7da3b76db61428d978ddfc6cb133a","abstract_canon_sha256":"1555e2b4375a9332917babd32ea066f7b3c563f8c89255ae196a20ea3b0ebcf1"},"schema_version":"1.0"},"canonical_sha256":"091dc031724f5c27035e2dd1064d617d42ba623472a0214ab98df4f165e959a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:27.535738Z","signature_b64":"2ghe8CzhIHyP1lEptyd30vlge68Hwa1lbUeQb33CIrrho1eJbuO2xFuH6iylinM3uyXwzLmZAecjfzIb/3OiCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"091dc031724f5c27035e2dd1064d617d42ba623472a0214ab98df4f165e959a0","last_reissued_at":"2026-05-18T00:20:27.535157Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:27.535157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.07257","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":"2026-05-18T00:20:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LkrtAYROAaoh0ZxDAdHR4oFNnrsJLexCikPRmXLWS8O0ZuYYHHeCf/+O5m+4EAVcCYk/Zk1swYZMOupuhT3QAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T03:53:53.461256Z"},"content_sha256":"98d687eb9a96d81ce39f0170bbdd1c2dc2132acbb73b67996f114d3b8adfdf4a","schema_version":"1.0","event_id":"sha256:98d687eb9a96d81ce39f0170bbdd1c2dc2132acbb73b67996f114d3b8adfdf4a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:BEO4AMLSJ5OCOA26FXIQMTLBPV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Further remarks on liftings of crossed modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CT","authors_text":"Tun\\c{c}ar \\c{S}ahan","submitted_at":"2017-04-24T14:39:07Z","abstract_excerpt":"In this paper we define the notion of pullback lifting of a lifting crossed module over a crossed module morphism and interpret this notion in the category of group-groupoid actions as pullback action. Moreover, we give a criterion for the lifting of homotopic crossed module morphisms to be homotopic, which will be called homotopy lifting property for crossed module morphisms. Finally, we investigate some properties of derivations of lifting crossed modules according to base crossed module derivations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07257","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":"2026-05-18T00:20:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZLEE8zfJLXxeB9Y8KJe3DTOPJLPOCkPsxDpm+aKgUFNdso4II5w4s7v2C7Rbc23khwX0NsfOa9UE0bJEEBZIAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T03:53:53.461629Z"},"content_sha256":"af09c6cc2d00dcbb968ddc93bfcf21ab49cffb2963caea752f9706b34c832940","schema_version":"1.0","event_id":"sha256:af09c6cc2d00dcbb968ddc93bfcf21ab49cffb2963caea752f9706b34c832940"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BEO4AMLSJ5OCOA26FXIQMTLBPV/bundle.json","state_url":"https://pith.science/pith/BEO4AMLSJ5OCOA26FXIQMTLBPV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BEO4AMLSJ5OCOA26FXIQMTLBPV/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":"2026-06-25T03:53:53Z","links":{"resolver":"https://pith.science/pith/BEO4AMLSJ5OCOA26FXIQMTLBPV","bundle":"https://pith.science/pith/BEO4AMLSJ5OCOA26FXIQMTLBPV/bundle.json","state":"https://pith.science/pith/BEO4AMLSJ5OCOA26FXIQMTLBPV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BEO4AMLSJ5OCOA26FXIQMTLBPV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BEO4AMLSJ5OCOA26FXIQMTLBPV","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":"1555e2b4375a9332917babd32ea066f7b3c563f8c89255ae196a20ea3b0ebcf1","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2017-04-24T14:39:07Z","title_canon_sha256":"3404332046f8abbe4036086894531226f3f7da3b76db61428d978ddfc6cb133a"},"schema_version":"1.0","source":{"id":"1704.07257","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.07257","created_at":"2026-05-18T00:20:27Z"},{"alias_kind":"arxiv_version","alias_value":"1704.07257v1","created_at":"2026-05-18T00:20:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.07257","created_at":"2026-05-18T00:20:27Z"},{"alias_kind":"pith_short_12","alias_value":"BEO4AMLSJ5OC","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BEO4AMLSJ5OCOA26","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BEO4AMLS","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:af09c6cc2d00dcbb968ddc93bfcf21ab49cffb2963caea752f9706b34c832940","target":"graph","created_at":"2026-05-18T00:20:27Z","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":"In this paper we define the notion of pullback lifting of a lifting crossed module over a crossed module morphism and interpret this notion in the category of group-groupoid actions as pullback action. Moreover, we give a criterion for the lifting of homotopic crossed module morphisms to be homotopic, which will be called homotopy lifting property for crossed module morphisms. Finally, we investigate some properties of derivations of lifting crossed modules according to base crossed module derivations.","authors_text":"Tun\\c{c}ar \\c{S}ahan","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2017-04-24T14:39:07Z","title":"Further remarks on liftings of crossed modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07257","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":"sha256:98d687eb9a96d81ce39f0170bbdd1c2dc2132acbb73b67996f114d3b8adfdf4a","target":"record","created_at":"2026-05-18T00:20:27Z","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":"1555e2b4375a9332917babd32ea066f7b3c563f8c89255ae196a20ea3b0ebcf1","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2017-04-24T14:39:07Z","title_canon_sha256":"3404332046f8abbe4036086894531226f3f7da3b76db61428d978ddfc6cb133a"},"schema_version":"1.0","source":{"id":"1704.07257","kind":"arxiv","version":1}},"canonical_sha256":"091dc031724f5c27035e2dd1064d617d42ba623472a0214ab98df4f165e959a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"091dc031724f5c27035e2dd1064d617d42ba623472a0214ab98df4f165e959a0","first_computed_at":"2026-05-18T00:20:27.535157Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:27.535157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2ghe8CzhIHyP1lEptyd30vlge68Hwa1lbUeQb33CIrrho1eJbuO2xFuH6iylinM3uyXwzLmZAecjfzIb/3OiCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:27.535738Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.07257","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:98d687eb9a96d81ce39f0170bbdd1c2dc2132acbb73b67996f114d3b8adfdf4a","sha256:af09c6cc2d00dcbb968ddc93bfcf21ab49cffb2963caea752f9706b34c832940"],"state_sha256":"b5c14d58341f5b2568222af122b94b657a03bdea7a09805abd6d7dbb77427efb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IwH5imp69TMR6jKOmjHx3/6MDzweWFuMyhTR80WCAcV9JQvX2J4QzJJfa9ZX3cQVpHEN7JTJDpEPOLfGeAg0BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T03:53:53.463594Z","bundle_sha256":"d0d751d5c7a92d32d7bf89acbaaf7017f07048fd77123d5868310c8515479c76"}}