{"paper":{"title":"Fa\\`a di Bruno for operads and internal algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.CT","authors_text":"Joachim Kock, Mark Weber","submitted_at":"2016-09-12T05:55:49Z","abstract_excerpt":"For any coloured operad R, we prove a Fa\\`a di Bruno formula for the `connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Fa\\`a di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faa di Bruno formula for P-trees of G\\'alvez--Kock--Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following G\\'alvez--Kock--Tonks, we work at the objective level of groupoid slices, hence all proofs are `bijectiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.03276","kind":"arxiv","version":1},"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"}