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Confluence laws and Hopf-Borel type theorem for operads

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Emily Burgunder (IMT), Bérénice Delcroix-Oger (IRIF)

Combinatorics · Submitted 5 Jan 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This enables us to show that for any operadsP and Q having the same underlying S-module, there exists a confluencelaw $\alpha$ such that any conilpotent P coQ-bialgebra satisfying $\alpha$ is free andcofree over its primitive elements.

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arXiv ID1701.01323 (version 3)
Pith IDDH6YADHD, Pith's permanent record of this paper and its review
CitationB\'er\'enice Delcroix-Oger (IRIF), Emily Burgunder (IMT). "Confluence laws and Hopf-Borel type theorem for operads." arXiv:1701.01323 (2017). https://pith.science/paper/DH6YADHD
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,073 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

In 2008, Loday shed light on the existence of Hopf-Boreltheorems for operads. Using the vocabulary of category theory, Livernet,Mesablishvili and Wisbauer extended such theorems to monads. In bothcases, the reasoning was to start from a mixed distributive law andthen to prove that it induces an isomorphism of species to finally geta rigidity theorem. Our reasoning goes here backward: we prove thatfrom an isomorphism of species one can get what we called a confluencelaw, which generalises mixed distributive laws, and that it is enough toobtain a rigidity theorem. This enables us to show that for any operadsP and Q having the same underlying S-module, there exists a confluencelaw $\alpha$ such that any conilpotent P coQ-bialgebra satisfying $\alpha$ is free andcofree over its primitive elements. Our reasoning permits us to generatemany new examples, while recovering the known ones by consideringdual relations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In 2008, Loday shed light on the existence of Hopf-Boreltheorems for operads. background 1.00 0.03 0.02 asserted 0.99
s1 Using the vocabulary of category theory, Livernet,Mesablishvili and Wisbauer extended such theorems to monads. background 1.00 0.03 0.02 asserted 1.00
s2 In bothcases, the reasoning was to start from a mixed distributive law andthen to prove that it induces an isomorphism of species to finally geta rigidity theorem. background 0.93 0.05 0.04 asserted 1.00
s3 Our reasoning goes here backward: we prove thatfrom an isomorphism of species one can get what we called a confluencelaw, which generalises mixed distributive laws, and that it is enough toobtain a rigidity theorem. contribution 0.59 0.95 0.22 asserted 1.00
s4 This enables us to show that for any operadsP and Q having the same underlying S-module, there exists a confluencelaw $\alpha$ such that any conilpotent P coQ-bialgebra satisfying $\alpha$ is free andcofree over its primitive elements. result 0.98 0.95 0.78 asserted 1.00
s5 Our reasoning permits us to generatemany new examples, while recovering the known ones by consideringdual relations. interpretation 0.44 0.90 0.08 asserted 0.90

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In 2008, Loday shed light on the existence of Hopf-Boreltheorems for operads. Using the vocabulary of category theory, Livernet,Mesablishvili and Wisbauer extended such theorems to monads. In bothcases, the reasoning was to start from a mixed distributive law andthen to prove that it induces an isomorphism of species to finally geta rigidity theorem. Our reasoning goes here backward: we prove thatfrom an isomorphism of species one can get what we called a confluencelaw, which generalises mixed distributive laws, and that it is enough toobtain a rigidity theorem. This enables us to show that for any operadsP and Q having the same underlying S-module, there exists a confluencelaw $\alpha$ such that any conilpotent P coQ-bialgebra satisfying $\alpha$ is free andcofree over its primitive elements. Our reasoning permits us to generatemany new examples, while recovering the known ones by consideringdual relations.

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