Cofree Compositions of Coalgebras Stefan Forcey, Aaron Lauve, Frank Sottile
Cofree compositions of coalgebras Stefan Forcey, Aaron Lauve, Frank Sottile To cite this version: Stefan Forcey, Aaron Lauve, Frank Sottile. Cofree compositions of coalgebras. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.363-374. hal-01215100 HAL Id: hal-01215100 https://hal.inria.fr/hal-01215100 Submitted on 13 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2011, Reykjav´ık, Iceland DMTCS proc. AO, 2011, 363–374 Cofree compositions of coalgebras Stefan Forcey1, Aaron Lauve2y and Frank Sottile3z 1Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH, USA 2Department of Mathematics, Loyola University of Chicago, Chicago, IL, USA 3Department of Mathematics, Texas A&M University, College Station, TX, USA 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.
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