{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:DLKNHOYDLUNINOH5BBLERVHB5Y","short_pith_number":"pith:DLKNHOYD","canonical_record":{"source":{"id":"1007.3868","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-07-22T12:30:38Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"06f6a0d4e37954e1d029d80a3b6fc8255b958f6505a2213a59fcb02817e3cc36","abstract_canon_sha256":"60f37cc2de97f19a6e8fd2cc62b8e1e8d8b8f6df77242a2fe60321d98896763d"},"schema_version":"1.0"},"canonical_sha256":"1ad4d3bb035d1a86b8fd085648d4e1ee3f2324e3a286c3cbf825d78778951b1b","source":{"kind":"arxiv","id":"1007.3868","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.3868","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1007.3868v3","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.3868","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"DLKNHOYDLUNI","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"DLKNHOYDLUNINOH5","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"DLKNHOYD","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:DLKNHOYDLUNINOH5BBLERVHB5Y","target":"record","payload":{"canonical_record":{"source":{"id":"1007.3868","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-07-22T12:30:38Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"06f6a0d4e37954e1d029d80a3b6fc8255b958f6505a2213a59fcb02817e3cc36","abstract_canon_sha256":"60f37cc2de97f19a6e8fd2cc62b8e1e8d8b8f6df77242a2fe60321d98896763d"},"schema_version":"1.0"},"canonical_sha256":"1ad4d3bb035d1a86b8fd085648d4e1ee3f2324e3a286c3cbf825d78778951b1b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:49.761213Z","signature_b64":"qn0UyWSi0vLvjAt1a/zdn3OcvDrNIp2rIllgJVSICExXU4GaXO7bzOkt+L+O7axSGRfv5MdgmOdvm1pozxySBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ad4d3bb035d1a86b8fd085648d4e1ee3f2324e3a286c3cbf825d78778951b1b","last_reissued_at":"2026-05-18T04:25:49.760464Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:49.760464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.3868","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":"2026-05-18T04:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"//zAS5cOU+QWfT6d65KwzPRlTPGXgOBg6MHxIY6qAimgWcffbwl2zW7jy/mHjZkWm2XP55im+0AZq8s+ECOECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:39:33.768240Z"},"content_sha256":"aee4185359e3d7664cff8ba520e493c1da5b660ebe1ad551a963fb468003dc95","schema_version":"1.0","event_id":"sha256:aee4185359e3d7664cff8ba520e493c1da5b660ebe1ad551a963fb468003dc95"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:DLKNHOYDLUNINOH5BBLERVHB5Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Euler Characteristics of Categories and Homotopy Colimits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Roman Sauer, Thomas M. Fiore, Wolfgang L\\\"uck","submitted_at":"2010-07-22T12:30:38Z","abstract_excerpt":"In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of I-indexed categories where I is any small category admitting a finite I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass--Serre graphs of groups to highe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.3868","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":""},"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":"2026-05-18T04:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qcNfBhexiRc4yQcqWdF3UiuSc0nS6BTfHIXDyt5iWJu8q/LAHhiguG1NMIbxlRxCEwy5KzORdfwsQk8wokj4Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:39:33.768596Z"},"content_sha256":"ac38534ab990f8e2a860f1ed1f0414e87e29302a61312980e99441d80b56864a","schema_version":"1.0","event_id":"sha256:ac38534ab990f8e2a860f1ed1f0414e87e29302a61312980e99441d80b56864a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DLKNHOYDLUNINOH5BBLERVHB5Y/bundle.json","state_url":"https://pith.science/pith/DLKNHOYDLUNINOH5BBLERVHB5Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DLKNHOYDLUNINOH5BBLERVHB5Y/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":"2026-06-03T16:39:33Z","links":{"resolver":"https://pith.science/pith/DLKNHOYDLUNINOH5BBLERVHB5Y","bundle":"https://pith.science/pith/DLKNHOYDLUNINOH5BBLERVHB5Y/bundle.json","state":"https://pith.science/pith/DLKNHOYDLUNINOH5BBLERVHB5Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DLKNHOYDLUNINOH5BBLERVHB5Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:DLKNHOYDLUNINOH5BBLERVHB5Y","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":"60f37cc2de97f19a6e8fd2cc62b8e1e8d8b8f6df77242a2fe60321d98896763d","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-07-22T12:30:38Z","title_canon_sha256":"06f6a0d4e37954e1d029d80a3b6fc8255b958f6505a2213a59fcb02817e3cc36"},"schema_version":"1.0","source":{"id":"1007.3868","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.3868","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1007.3868v3","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.3868","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"DLKNHOYDLUNI","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"DLKNHOYDLUNINOH5","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"DLKNHOYD","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:ac38534ab990f8e2a860f1ed1f0414e87e29302a61312980e99441d80b56864a","target":"graph","created_at":"2026-05-18T04:25:49Z","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 a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of I-indexed categories where I is any small category admitting a finite I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass--Serre graphs of groups to highe","authors_text":"Roman Sauer, Thomas M. Fiore, Wolfgang L\\\"uck","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-07-22T12:30:38Z","title":"Euler Characteristics of Categories and Homotopy Colimits"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.3868","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:aee4185359e3d7664cff8ba520e493c1da5b660ebe1ad551a963fb468003dc95","target":"record","created_at":"2026-05-18T04:25:49Z","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":"60f37cc2de97f19a6e8fd2cc62b8e1e8d8b8f6df77242a2fe60321d98896763d","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-07-22T12:30:38Z","title_canon_sha256":"06f6a0d4e37954e1d029d80a3b6fc8255b958f6505a2213a59fcb02817e3cc36"},"schema_version":"1.0","source":{"id":"1007.3868","kind":"arxiv","version":3}},"canonical_sha256":"1ad4d3bb035d1a86b8fd085648d4e1ee3f2324e3a286c3cbf825d78778951b1b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1ad4d3bb035d1a86b8fd085648d4e1ee3f2324e3a286c3cbf825d78778951b1b","first_computed_at":"2026-05-18T04:25:49.760464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:49.760464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qn0UyWSi0vLvjAt1a/zdn3OcvDrNIp2rIllgJVSICExXU4GaXO7bzOkt+L+O7axSGRfv5MdgmOdvm1pozxySBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:49.761213Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.3868","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aee4185359e3d7664cff8ba520e493c1da5b660ebe1ad551a963fb468003dc95","sha256:ac38534ab990f8e2a860f1ed1f0414e87e29302a61312980e99441d80b56864a"],"state_sha256":"50cae7d3e66a540077c420e05147995592a92a1a1882af0f9c9c74baa743ff34"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SQ3kBxzUh/m7iKyT5MJ9iPiXI4WTkByY9v1CihxG5oxDp4T49EcbIJIWr3bO6nb5v4PVCXTm4493EgWzM/f1AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T16:39:33.770570Z","bundle_sha256":"e5cdef09d843b87aa147283384a6a44623cf57b8c1ac4f3428840b575d624e39"}}