{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:BDFPYRMJWTEBN4TH44PDFKJXY3","short_pith_number":"pith:BDFPYRMJ","schema_version":"1.0","canonical_sha256":"08cafc4589b4c816f267e71e32a937c6cbfe942d985e6800b1192a55b1e5159b","source":{"kind":"arxiv","id":"2104.02729","version":4},"attestation_state":"computed","paper":{"title":"Configuration spaces of clusters as $E_d$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Florian Kranhold","submitted_at":"2021-04-06T18:00:14Z","abstract_excerpt":"It is a classical result that configuration spaces of labelled particles in $\\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular $E_1$-algebras, and the other one uses"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2104.02729","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2021-04-06T18:00:14Z","cross_cats_sorted":[],"title_canon_sha256":"9c94f1fff9818077e1915e0da7cd611193e2d3d00c3f5357d33cf3c664605275","abstract_canon_sha256":"ad0037a24ed1f07bea623ef903128cd0938c0716e74037eb891c1f767ae1c351"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:24:52.098404Z","signature_b64":"nOeIfD/1wjyNgSai525wzsywhC9/EPdYsY/ipLXAYS11wm5gKIbHKpsPIOc1lg/SMDd3poIqi/dHOpE8jua6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08cafc4589b4c816f267e71e32a937c6cbfe942d985e6800b1192a55b1e5159b","last_reissued_at":"2026-07-05T08:24:52.097975Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:24:52.097975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Configuration spaces of clusters as $E_d$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Florian Kranhold","submitted_at":"2021-04-06T18:00:14Z","abstract_excerpt":"It is a classical result that configuration spaces of labelled particles in $\\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular $E_1$-algebras, and the other one uses"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.02729","kind":"arxiv","version":4},"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/2104.02729/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2104.02729","created_at":"2026-07-05T08:24:52.098035+00:00"},{"alias_kind":"arxiv_version","alias_value":"2104.02729v4","created_at":"2026-07-05T08:24:52.098035+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.02729","created_at":"2026-07-05T08:24:52.098035+00:00"},{"alias_kind":"pith_short_12","alias_value":"BDFPYRMJWTEB","created_at":"2026-07-05T08:24:52.098035+00:00"},{"alias_kind":"pith_short_16","alias_value":"BDFPYRMJWTEBN4TH","created_at":"2026-07-05T08:24:52.098035+00:00"},{"alias_kind":"pith_short_8","alias_value":"BDFPYRMJ","created_at":"2026-07-05T08:24:52.098035+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.01128","citing_title":"Representation stability in the (co)homology of vertical configuration spaces","ref_index":25,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3","json":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3.json","graph_json":"https://pith.science/api/pith-number/BDFPYRMJWTEBN4TH44PDFKJXY3/graph.json","events_json":"https://pith.science/api/pith-number/BDFPYRMJWTEBN4TH44PDFKJXY3/events.json","paper":"https://pith.science/paper/BDFPYRMJ"},"agent_actions":{"view_html":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3","download_json":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3.json","view_paper":"https://pith.science/paper/BDFPYRMJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2104.02729&json=true","fetch_graph":"https://pith.science/api/pith-number/BDFPYRMJWTEBN4TH44PDFKJXY3/graph.json","fetch_events":"https://pith.science/api/pith-number/BDFPYRMJWTEBN4TH44PDFKJXY3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/action/storage_attestation","attest_author":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/action/author_attestation","sign_citation":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/action/citation_signature","submit_replication":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/action/replication_record"}},"created_at":"2026-07-05T08:24:52.098035+00:00","updated_at":"2026-07-05T08:24:52.098035+00:00"}