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The collection $\\overline{\\mathcal{M}}=\\{\\overline{\\mathcal{M}}_{0,n+1}\\}_{n\\geq 2}$ forms an operad in the category of complex projective varieties; its homology $Hycom= H_*(\\overline{\\mathcal{M}})$ is called the Hypercommutative operad. In this paper we construct a chain model for the hypercommutative operad, i.e. an operad of chain complexes $C_*^{dual}(\\overline{\\mathcal{M}})$ which is weakly equivalent to the operad of singular chains $C_*(\\overline{\\mathcal{M}})$. We prove tha"},"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":"2412.03474","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-12-04T17:02:17Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"8d2e5e61ba9dd6b638b6586220eaf83fc91d6c8b54d4659ae591b9073e06506a","abstract_canon_sha256":"25f96a669f9fc9e5b53e5203b475508d7d856dda0d69ff8a13ab097a42045106"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:44:30.116121Z","signature_b64":"pXbabgKulRr1AmZY4x3+92S4KxfLkX5u0G8Swhb53XzLSV4er3q3/h6PN1BUmQ+ftgPc33r3+DB3xFJxz0jvBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf6fcdd02aba7795b4b9612d6d14f1810117fc8dc65bbe63b1722011caaf779b","last_reissued_at":"2026-07-05T09:44:30.115640Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:44:30.115640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chain level Koszul duality between the Gravity and Hypercommutative operads","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AT","authors_text":"Paolo Salvatore, Tommaso Rossi","submitted_at":"2024-12-04T17:02:17Z","abstract_excerpt":"Let $\\overline{\\mathcal{M}}_{0,n+1}$ be the moduli space of genus zero stable curves with $(n+1)$-marked points. The collection $\\overline{\\mathcal{M}}=\\{\\overline{\\mathcal{M}}_{0,n+1}\\}_{n\\geq 2}$ forms an operad in the category of complex projective varieties; its homology $Hycom= H_*(\\overline{\\mathcal{M}})$ is called the Hypercommutative operad. In this paper we construct a chain model for the hypercommutative operad, i.e. an operad of chain complexes $C_*^{dual}(\\overline{\\mathcal{M}})$ which is weakly equivalent to the operad of singular chains $C_*(\\overline{\\mathcal{M}})$. 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