{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:7DXJ7BCGZ5CUIOLZXZJVZOK6I5","short_pith_number":"pith:7DXJ7BCG","canonical_record":{"source":{"id":"2010.15722","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-10-29T16:11:18Z","cross_cats_sorted":["math.AG","math.AT"],"title_canon_sha256":"ae9be7a06de7e6a2e7936460bb2eaacffa9ca4367d1cce14b4ebc8e3e4dfe9c8","abstract_canon_sha256":"5d7154a74d614de48a56173193c0aa49014a9823840f585e9e13523ac48f05b4"},"schema_version":"1.0"},"canonical_sha256":"f8ee9f8446cf45443979be535cb95e4773bbb16bb95caf05f78f52ad073d2703","source":{"kind":"arxiv","id":"2010.15722","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.15722","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"arxiv_version","alias_value":"2010.15722v3","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.15722","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"pith_short_12","alias_value":"7DXJ7BCGZ5CU","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"pith_short_16","alias_value":"7DXJ7BCGZ5CUIOLZ","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"pith_short_8","alias_value":"7DXJ7BCG","created_at":"2026-07-05T03:35:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:7DXJ7BCGZ5CUIOLZXZJVZOK6I5","target":"record","payload":{"canonical_record":{"source":{"id":"2010.15722","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-10-29T16:11:18Z","cross_cats_sorted":["math.AG","math.AT"],"title_canon_sha256":"ae9be7a06de7e6a2e7936460bb2eaacffa9ca4367d1cce14b4ebc8e3e4dfe9c8","abstract_canon_sha256":"5d7154a74d614de48a56173193c0aa49014a9823840f585e9e13523ac48f05b4"},"schema_version":"1.0"},"canonical_sha256":"f8ee9f8446cf45443979be535cb95e4773bbb16bb95caf05f78f52ad073d2703","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:35:26.674296Z","signature_b64":"fuBntK7wihHvwnggqWd9A3G+6kDQl+B29mGVKrMzGZ4w49vVV6hncZbAn3iJgHLB61XUMdyd3zGtAbP1ioXLBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8ee9f8446cf45443979be535cb95e4773bbb16bb95caf05f78f52ad073d2703","last_reissued_at":"2026-07-05T03:35:26.673891Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:35:26.673891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2010.15722","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:35:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j58/fWjU+Cegz5By0nzDvyZxClDyu9a2OP/9V7lK/jgoDqJuOowuI6wVTw+mtOZy5H+C57cpsaMjKvgCvET3AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T22:37:28.536848Z"},"content_sha256":"64031a8a91fdfe8e5488629d524d8e4d2d6237193258ba660680ef32aba045ba","schema_version":"1.0","event_id":"sha256:64031a8a91fdfe8e5488629d524d8e4d2d6237193258ba660680ef32aba045ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:7DXJ7BCGZ5CUIOLZXZJVZOK6I5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On distributivity in higher algebra I: The universal property of bispans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.AT"],"primary_cat":"math.CT","authors_text":"Elden Elmanto, Rune Haugseng","submitted_at":"2020-10-29T16:11:18Z","abstract_excerpt":"Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\\infty$-)categories of spans (or correspondences). In this paper we study the more complicated setup where we have two pushforwards (an \"additive\" and a \"multiplicative\" one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\\infty$-cat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.15722","kind":"arxiv","version":3},"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/2010.15722/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":"2026-07-05T03:35:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fFhlKWmrZv4Jzt/iQRrViFUnWLmykpCP4cBeVCcaZ7yPlwMzjYX9FNuLalm4jXq3tlQq1BpkpnmDwnDkdxpTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T22:37:28.537337Z"},"content_sha256":"c39549414c551c9946f6f202a90d59d94aa41ce49bc3756698851fe8492979d3","schema_version":"1.0","event_id":"sha256:c39549414c551c9946f6f202a90d59d94aa41ce49bc3756698851fe8492979d3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7DXJ7BCGZ5CUIOLZXZJVZOK6I5/bundle.json","state_url":"https://pith.science/pith/7DXJ7BCGZ5CUIOLZXZJVZOK6I5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7DXJ7BCGZ5CUIOLZXZJVZOK6I5/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":"2026-08-12T22:37:28Z","links":{"resolver":"https://pith.science/pith/7DXJ7BCGZ5CUIOLZXZJVZOK6I5","bundle":"https://pith.science/pith/7DXJ7BCGZ5CUIOLZXZJVZOK6I5/bundle.json","state":"https://pith.science/pith/7DXJ7BCGZ5CUIOLZXZJVZOK6I5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7DXJ7BCGZ5CUIOLZXZJVZOK6I5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:7DXJ7BCGZ5CUIOLZXZJVZOK6I5","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":"5d7154a74d614de48a56173193c0aa49014a9823840f585e9e13523ac48f05b4","cross_cats_sorted":["math.AG","math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-10-29T16:11:18Z","title_canon_sha256":"ae9be7a06de7e6a2e7936460bb2eaacffa9ca4367d1cce14b4ebc8e3e4dfe9c8"},"schema_version":"1.0","source":{"id":"2010.15722","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.15722","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"arxiv_version","alias_value":"2010.15722v3","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.15722","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"pith_short_12","alias_value":"7DXJ7BCGZ5CU","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"pith_short_16","alias_value":"7DXJ7BCGZ5CUIOLZ","created_at":"2026-07-05T03:35:26Z"},{"alias_kind":"pith_short_8","alias_value":"7DXJ7BCG","created_at":"2026-07-05T03:35:26Z"}],"graph_snapshots":[{"event_id":"sha256:c39549414c551c9946f6f202a90d59d94aa41ce49bc3756698851fe8492979d3","target":"graph","created_at":"2026-07-05T03:35:26Z","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/2010.15722/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\\infty$-)categories of spans (or correspondences). In this paper we study the more complicated setup where we have two pushforwards (an \"additive\" and a \"multiplicative\" one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\\infty$-cat","authors_text":"Elden Elmanto, Rune Haugseng","cross_cats":["math.AG","math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-10-29T16:11:18Z","title":"On distributivity in higher algebra I: The universal property of bispans"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.15722","kind":"arxiv","version":3},"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":"sha256:64031a8a91fdfe8e5488629d524d8e4d2d6237193258ba660680ef32aba045ba","target":"record","created_at":"2026-07-05T03:35:26Z","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":"5d7154a74d614de48a56173193c0aa49014a9823840f585e9e13523ac48f05b4","cross_cats_sorted":["math.AG","math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-10-29T16:11:18Z","title_canon_sha256":"ae9be7a06de7e6a2e7936460bb2eaacffa9ca4367d1cce14b4ebc8e3e4dfe9c8"},"schema_version":"1.0","source":{"id":"2010.15722","kind":"arxiv","version":3}},"canonical_sha256":"f8ee9f8446cf45443979be535cb95e4773bbb16bb95caf05f78f52ad073d2703","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8ee9f8446cf45443979be535cb95e4773bbb16bb95caf05f78f52ad073d2703","first_computed_at":"2026-07-05T03:35:26.673891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:35:26.673891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fuBntK7wihHvwnggqWd9A3G+6kDQl+B29mGVKrMzGZ4w49vVV6hncZbAn3iJgHLB61XUMdyd3zGtAbP1ioXLBg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:35:26.674296Z","signed_message":"canonical_sha256_bytes"},"source_id":"2010.15722","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64031a8a91fdfe8e5488629d524d8e4d2d6237193258ba660680ef32aba045ba","sha256:c39549414c551c9946f6f202a90d59d94aa41ce49bc3756698851fe8492979d3"],"state_sha256":"dd5be280811f57bb547d4cb5a53b50c881e734dab89e7590490c969021e5aa88"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xrhPbIffsKmXu+JU/H2NF6wX+7AU7AwEDlaypBDOXLUBDLqNsRS3V3/8QdHvF+XL3ILuWuECDrfaldC0xoPnAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T22:37:28.540669Z","bundle_sha256":"b6fbddf9fa82c22af3a36f8189ac782ed1254cdd2c1f91c7529f40aa42dc66c3"}}