{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:IHPIMKGE2NVW3JSNLAUZV7U3RW","short_pith_number":"pith:IHPIMKGE","canonical_record":{"source":{"id":"2109.07816","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.GN","submitted_at":"2021-09-16T08:59:19Z","cross_cats_sorted":[],"title_canon_sha256":"960ae1c82939ac8cd10ac381f8e284c1692afba7bf6235d4ae9827cc1e31ea48","abstract_canon_sha256":"bd9e6ce03b65d062992f6ed0b79ba9fc7d48c381cbcc4e9af1b9da2b5ea0c7f5"},"schema_version":"1.0"},"canonical_sha256":"41de8628c4d36b6da64d58299afe9b8d98f2164f8911df170d51f6973df59dfe","source":{"kind":"arxiv","id":"2109.07816","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.07816","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"arxiv_version","alias_value":"2109.07816v1","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.07816","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"pith_short_12","alias_value":"IHPIMKGE2NVW","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"pith_short_16","alias_value":"IHPIMKGE2NVW3JSN","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"pith_short_8","alias_value":"IHPIMKGE","created_at":"2026-07-05T03:15:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:IHPIMKGE2NVW3JSNLAUZV7U3RW","target":"record","payload":{"canonical_record":{"source":{"id":"2109.07816","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.GN","submitted_at":"2021-09-16T08:59:19Z","cross_cats_sorted":[],"title_canon_sha256":"960ae1c82939ac8cd10ac381f8e284c1692afba7bf6235d4ae9827cc1e31ea48","abstract_canon_sha256":"bd9e6ce03b65d062992f6ed0b79ba9fc7d48c381cbcc4e9af1b9da2b5ea0c7f5"},"schema_version":"1.0"},"canonical_sha256":"41de8628c4d36b6da64d58299afe9b8d98f2164f8911df170d51f6973df59dfe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:15:01.521469Z","signature_b64":"mPzDPu0GC1Hxs5uFsIn4plBTDHLkFutwDrEG6dabGR1EGwr3w5FnOga0tnGzDpeIVU6IWOjacs7/s+uTg+jxAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41de8628c4d36b6da64d58299afe9b8d98f2164f8911df170d51f6973df59dfe","last_reissued_at":"2026-07-05T03:15:01.521024Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:15:01.521024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.07816","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":"2026-07-05T03:15:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QzjpQQS1sPxdlN9Ukkk38bpq6JSUpiunMqYFhHbydwHOCLWilabySwwsF4haiHA7synvEaLAazmYFd7eQTxBCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T04:48:31.950450Z"},"content_sha256":"2b414c3f736163418cf383bb44afa24a25b95f5a9b336d2e7aea5b41b7dfcd35","schema_version":"1.0","event_id":"sha256:2b414c3f736163418cf383bb44afa24a25b95f5a9b336d2e7aea5b41b7dfcd35"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:IHPIMKGE2NVW3JSNLAUZV7U3RW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Condensed Mathematics: The internal Hom of condensed sets and condensed abelian groups and a prismatic construction of the real numbers","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Rodrigo Marlasca Aparicio","submitted_at":"2021-09-16T08:59:19Z","abstract_excerpt":"We give an overview of the basic definitions of condensed categories, as well as the internal Hom of condensed abelian groups. We give a construction for the internal Hom of condensed sets and apply it to obtain a new proof of a theorem of Clausen and Scholze. Finally, we give a detailed account of a construction of the real numbers from discrete spaces, which is an intermediate step of a theorem by Clausen and Scholze."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.07816","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.07816/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":"2026-07-05T03:15:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SPzRW3aQAYqomyeqZ4rXnES5MJ7BKYIYqFMCmoezK3wW5Aq92F7jQH2OrO+qRU5hwGVpLHnyRJB/tG7pZfgoDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T04:48:31.951104Z"},"content_sha256":"66349b3400707bb18746f079a2461f386e83e621cba86055d4bc511e15d981d9","schema_version":"1.0","event_id":"sha256:66349b3400707bb18746f079a2461f386e83e621cba86055d4bc511e15d981d9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IHPIMKGE2NVW3JSNLAUZV7U3RW/bundle.json","state_url":"https://pith.science/pith/IHPIMKGE2NVW3JSNLAUZV7U3RW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IHPIMKGE2NVW3JSNLAUZV7U3RW/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":"2026-08-13T04:48:31Z","links":{"resolver":"https://pith.science/pith/IHPIMKGE2NVW3JSNLAUZV7U3RW","bundle":"https://pith.science/pith/IHPIMKGE2NVW3JSNLAUZV7U3RW/bundle.json","state":"https://pith.science/pith/IHPIMKGE2NVW3JSNLAUZV7U3RW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IHPIMKGE2NVW3JSNLAUZV7U3RW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:IHPIMKGE2NVW3JSNLAUZV7U3RW","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":"bd9e6ce03b65d062992f6ed0b79ba9fc7d48c381cbcc4e9af1b9da2b5ea0c7f5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.GN","submitted_at":"2021-09-16T08:59:19Z","title_canon_sha256":"960ae1c82939ac8cd10ac381f8e284c1692afba7bf6235d4ae9827cc1e31ea48"},"schema_version":"1.0","source":{"id":"2109.07816","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.07816","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"arxiv_version","alias_value":"2109.07816v1","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.07816","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"pith_short_12","alias_value":"IHPIMKGE2NVW","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"pith_short_16","alias_value":"IHPIMKGE2NVW3JSN","created_at":"2026-07-05T03:15:01Z"},{"alias_kind":"pith_short_8","alias_value":"IHPIMKGE","created_at":"2026-07-05T03:15:01Z"}],"graph_snapshots":[{"event_id":"sha256:66349b3400707bb18746f079a2461f386e83e621cba86055d4bc511e15d981d9","target":"graph","created_at":"2026-07-05T03:15:01Z","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/2109.07816/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give an overview of the basic definitions of condensed categories, as well as the internal Hom of condensed abelian groups. We give a construction for the internal Hom of condensed sets and apply it to obtain a new proof of a theorem of Clausen and Scholze. Finally, we give a detailed account of a construction of the real numbers from discrete spaces, which is an intermediate step of a theorem by Clausen and Scholze.","authors_text":"Rodrigo Marlasca Aparicio","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.GN","submitted_at":"2021-09-16T08:59:19Z","title":"Condensed Mathematics: The internal Hom of condensed sets and condensed abelian groups and a prismatic construction of the real numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.07816","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":"sha256:2b414c3f736163418cf383bb44afa24a25b95f5a9b336d2e7aea5b41b7dfcd35","target":"record","created_at":"2026-07-05T03:15:01Z","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":"bd9e6ce03b65d062992f6ed0b79ba9fc7d48c381cbcc4e9af1b9da2b5ea0c7f5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.GN","submitted_at":"2021-09-16T08:59:19Z","title_canon_sha256":"960ae1c82939ac8cd10ac381f8e284c1692afba7bf6235d4ae9827cc1e31ea48"},"schema_version":"1.0","source":{"id":"2109.07816","kind":"arxiv","version":1}},"canonical_sha256":"41de8628c4d36b6da64d58299afe9b8d98f2164f8911df170d51f6973df59dfe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"41de8628c4d36b6da64d58299afe9b8d98f2164f8911df170d51f6973df59dfe","first_computed_at":"2026-07-05T03:15:01.521024Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:15:01.521024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mPzDPu0GC1Hxs5uFsIn4plBTDHLkFutwDrEG6dabGR1EGwr3w5FnOga0tnGzDpeIVU6IWOjacs7/s+uTg+jxAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:15:01.521469Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.07816","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b414c3f736163418cf383bb44afa24a25b95f5a9b336d2e7aea5b41b7dfcd35","sha256:66349b3400707bb18746f079a2461f386e83e621cba86055d4bc511e15d981d9"],"state_sha256":"e7d7a78cdb342fd33cfef9769f02af6518162f6dc2d2067de5e9a1f187708959"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ihCpCc1qaoHoteteBhYrzLrE+BkjjfHqez5kAiPRAcF/j3Hf9my24THmEaw9nGNGXUdq9GylqRsqf8HLBwMOAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T04:48:31.960610Z","bundle_sha256":"673de8900ad94e20fedcdd2b6df4c6c0069dac174a229e4d5312290f4c6f13d1"}}