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Cofree compositions of coalgebras (extended abstract)

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Stefan Forcey, Aaron Lauve, Frank Sottile

Combinatorics · Submitted 18 Nov 2010

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

The authors' main claim We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra.

how it is stated · asserted 1.00 read the record →

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arXiv ID1011.4305 (version 2)
Pith IDOTQ6ZP6E, Pith's permanent record of this paper and its review
CitationAaron Lauve, Frank Sottile, Stefan Forcey. "Cofree compositions of coalgebras (extended abstract)." arXiv:1011.4305 (2010). https://pith.science/paper/OTQ6ZP6E
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,607 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes.

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 We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. contribution 0.97 0.92 0.04 asserted 1.00
s1 We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. result 0.97 0.96 0.73 asserted 1.00
s2 These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. result 0.80 0.75 0.10 asserted 0.99
s3 We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes. contribution 0.88 0.85 0.15 asserted 0.65

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

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sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper
  • We prove that the composition of two cofree coalgebras is again cofree, and we give sufficient conditions that ensure the composition is a one-sided Hopf algebra.

    Cofree compositions of coalgebras · 1.00

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