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This confirms a conjecture of Lurie. The proof of this theorem consists of two parts, which are of independent interest. We first show that the Goodwillie derivatives can be refined to a lax functor $\\partial_* \\co"},"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":"2410.20504","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-10-27T16:38:42Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"113964ea7d19759070cd7d340c5013e2a238a8fa23b09284de9d6b5335583f69","abstract_canon_sha256":"ca67c0096602f3442ab81303de8f59b7c0557f73f1db5fa0ee437bd4e8b1c2a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:26:44.540875Z","signature_b64":"61nSDBM1iJ6r0fdCvx+xZUYhFOEfR8nY9iMD2/2Vh9OsalGZCazL0RYNRrND1XX1u+gRkCJR+9Kq/YRMXDZCCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2c0742961c7fc319b65e2b5d2afaf89196b20f9fd42e7ea1b5d9fe2435ca25f","last_reissued_at":"2026-07-05T11:26:44.540404Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:26:44.540404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the chain rule in Goodwillie calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Max Blans, Thomas Blom","submitted_at":"2024-10-27T16:38:42Z","abstract_excerpt":"We prove a generalization of the Arone-Ching chain rule for Goodwillie derivatives by showing that for any pair of reduced finitary functors $F \\colon \\mathcal{D} \\to \\mathcal{E}$ and $G \\colon \\mathcal{C} \\to \\mathcal{D}$ between differentiable $\\infty$-categories, there is an equivalence $\\partial_*(FG) \\simeq \\partial_*F \\circ_{\\partial_*{\\mathrm{id}_{\\mathcal{D}}}} \\partial_*G$. This confirms a conjecture of Lurie. The proof of this theorem consists of two parts, which are of independent interest. 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