{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:JG3YUQIDHOKB6P4QRM3N2WDIK4","short_pith_number":"pith:JG3YUQID","canonical_record":{"source":{"id":"2409.20382","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-09-30T15:18:15Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"0ac1f3c32835aead0cc1d92919186c43ff46b190f688479f2b3dff9e6e1115f4","abstract_canon_sha256":"480499d0acd08505b046eae20f5338c7449b0b3df0424fb37823d46b20ff8393"},"schema_version":"1.0"},"canonical_sha256":"49b78a41033b941f3f908b36dd5868572faf5b94ed43ac0ba66106fa92cdcc54","source":{"kind":"arxiv","id":"2409.20382","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.20382","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"arxiv_version","alias_value":"2409.20382v2","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.20382","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"pith_short_12","alias_value":"JG3YUQIDHOKB","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"pith_short_16","alias_value":"JG3YUQIDHOKB6P4Q","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"pith_short_8","alias_value":"JG3YUQID","created_at":"2026-07-05T10:02:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:JG3YUQIDHOKB6P4QRM3N2WDIK4","target":"record","payload":{"canonical_record":{"source":{"id":"2409.20382","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-09-30T15:18:15Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"0ac1f3c32835aead0cc1d92919186c43ff46b190f688479f2b3dff9e6e1115f4","abstract_canon_sha256":"480499d0acd08505b046eae20f5338c7449b0b3df0424fb37823d46b20ff8393"},"schema_version":"1.0"},"canonical_sha256":"49b78a41033b941f3f908b36dd5868572faf5b94ed43ac0ba66106fa92cdcc54","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:02:06.867235Z","signature_b64":"0S7nl+pj9gNNJQbXILTXCf1K5xi5ifadidCUL11jpjNW5ziLGexLZarHHIfMN6kNkpevsD6amR9+J26BJR24BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49b78a41033b941f3f908b36dd5868572faf5b94ed43ac0ba66106fa92cdcc54","last_reissued_at":"2026-07-05T10:02:06.866767Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:02:06.866767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2409.20382","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:02:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C5DAhIyBDp/ct6DAHVtbHVsntiP3TKwKE2ybuNkY8lRUYlLu7Fz+ne/y36coLwi+HabSaLXKxnEgzxtKdoXPCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:44:14.180456Z"},"content_sha256":"9b993c6674ca751d838cece0ce314da52eaa70940bb31e2166118bf8b8a82fd6","schema_version":"1.0","event_id":"sha256:9b993c6674ca751d838cece0ce314da52eaa70940bb31e2166118bf8b8a82fd6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:JG3YUQIDHOKB6P4QRM3N2WDIK4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-representable six-functor formalisms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AG","authors_text":"Alessandro D'Angelo, Chirantan Chowdhury","submitted_at":"2024-09-30T15:18:15Z","abstract_excerpt":"In this article, we study the properties of motivic homotopy category $\\mathcal{SH}_{\\operatorname{ext}}(\\mathcal{X})$ developed by Chowdhury and Khan-Ravi for $\\mathcal{X}$ a Nis-loc Stack. In particular, we compare the above construction with Voevodsky's original construction using NisLoc topology. Using the techniques developed by Liu-Zheng and Mann's notion of $\\infty$-category of correspondences and abstract six-functor formalisms, we also extend the exceptional functors and extend properties like projection formula, base change and purity to the non-representable situation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.20382","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.20382/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:02:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"czyb8rgQCzGl8lv4rW+j2Eza06glrEcYuLrmC8qE4aVxrBFdcR8cra3Tuy2nxXM9pp/LN+MvAyj9gFS2mEnxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:44:14.181166Z"},"content_sha256":"c43e581a2298e8a69e469e996c3faccb9d62be72b3f4669ad1e5b7012c5ba2ed","schema_version":"1.0","event_id":"sha256:c43e581a2298e8a69e469e996c3faccb9d62be72b3f4669ad1e5b7012c5ba2ed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JG3YUQIDHOKB6P4QRM3N2WDIK4/bundle.json","state_url":"https://pith.science/pith/JG3YUQIDHOKB6P4QRM3N2WDIK4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JG3YUQIDHOKB6P4QRM3N2WDIK4/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":"2026-08-12T08:44:14Z","links":{"resolver":"https://pith.science/pith/JG3YUQIDHOKB6P4QRM3N2WDIK4","bundle":"https://pith.science/pith/JG3YUQIDHOKB6P4QRM3N2WDIK4/bundle.json","state":"https://pith.science/pith/JG3YUQIDHOKB6P4QRM3N2WDIK4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JG3YUQIDHOKB6P4QRM3N2WDIK4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:JG3YUQIDHOKB6P4QRM3N2WDIK4","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":"480499d0acd08505b046eae20f5338c7449b0b3df0424fb37823d46b20ff8393","cross_cats_sorted":["math.KT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-09-30T15:18:15Z","title_canon_sha256":"0ac1f3c32835aead0cc1d92919186c43ff46b190f688479f2b3dff9e6e1115f4"},"schema_version":"1.0","source":{"id":"2409.20382","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.20382","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"arxiv_version","alias_value":"2409.20382v2","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.20382","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"pith_short_12","alias_value":"JG3YUQIDHOKB","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"pith_short_16","alias_value":"JG3YUQIDHOKB6P4Q","created_at":"2026-07-05T10:02:06Z"},{"alias_kind":"pith_short_8","alias_value":"JG3YUQID","created_at":"2026-07-05T10:02:06Z"}],"graph_snapshots":[{"event_id":"sha256:c43e581a2298e8a69e469e996c3faccb9d62be72b3f4669ad1e5b7012c5ba2ed","target":"graph","created_at":"2026-07-05T10:02:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2409.20382/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, we study the properties of motivic homotopy category $\\mathcal{SH}_{\\operatorname{ext}}(\\mathcal{X})$ developed by Chowdhury and Khan-Ravi for $\\mathcal{X}$ a Nis-loc Stack. In particular, we compare the above construction with Voevodsky's original construction using NisLoc topology. Using the techniques developed by Liu-Zheng and Mann's notion of $\\infty$-category of correspondences and abstract six-functor formalisms, we also extend the exceptional functors and extend properties like projection formula, base change and purity to the non-representable situation.","authors_text":"Alessandro D'Angelo, Chirantan Chowdhury","cross_cats":["math.KT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-09-30T15:18:15Z","title":"Non-representable six-functor formalisms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.20382","kind":"arxiv","version":2},"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":"sha256:9b993c6674ca751d838cece0ce314da52eaa70940bb31e2166118bf8b8a82fd6","target":"record","created_at":"2026-07-05T10:02:06Z","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":"480499d0acd08505b046eae20f5338c7449b0b3df0424fb37823d46b20ff8393","cross_cats_sorted":["math.KT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-09-30T15:18:15Z","title_canon_sha256":"0ac1f3c32835aead0cc1d92919186c43ff46b190f688479f2b3dff9e6e1115f4"},"schema_version":"1.0","source":{"id":"2409.20382","kind":"arxiv","version":2}},"canonical_sha256":"49b78a41033b941f3f908b36dd5868572faf5b94ed43ac0ba66106fa92cdcc54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49b78a41033b941f3f908b36dd5868572faf5b94ed43ac0ba66106fa92cdcc54","first_computed_at":"2026-07-05T10:02:06.866767Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:02:06.866767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0S7nl+pj9gNNJQbXILTXCf1K5xi5ifadidCUL11jpjNW5ziLGexLZarHHIfMN6kNkpevsD6amR9+J26BJR24BA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:02:06.867235Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.20382","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b993c6674ca751d838cece0ce314da52eaa70940bb31e2166118bf8b8a82fd6","sha256:c43e581a2298e8a69e469e996c3faccb9d62be72b3f4669ad1e5b7012c5ba2ed"],"state_sha256":"39dac8b5ecb59ada8ee275423321ac70711ebe1d5d80bfbbd000dfc3497e06f3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yuy4WQofqs5ecys7qNBIIdCBqP6CE5qokVZqM+0dpqpBcqmxpdJElo/ExF9TYvtm0DwbrI6xnpkm+w/Ai6d8Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T08:44:14.188450Z","bundle_sha256":"82388449f15ceabf4c6053bb19a901c33ec04f935eb658a405320eba1957b590"}}