Annals of Combinatorics

, Volume 17, Issue 1, pp 105–130 | Cite as

Cofree Compositions of Coalgebras

  • Stefan Forcey
  • Aaron Lauve
  • Frank SottileEmail author


We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. 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. We show that these conditions are satisfied when one coalgebra is a graded Hopf operad \({\mathcal{D}}\) and the other is a connected graded coalgebra with coalgebra map to \({\mathcal{D}}\). We conclude by computing the primitive elements for compositions of coalgebras built on the vertices of multiplihedra, composihedra, and hypercubes.

Mathematics Subject Classification

05E05 16W30 18D50 


multiplihedron composihedron binary tree cofree coalgebra one-sided Hopf algebra operads species 


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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Theoretical and Applied MathematicsThe University of AkronAkronUSA
  2. 2.Department of MathematicsLoyola University of ChicagoChicagoUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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