2014/09/16 by Matthieu Anel, M. Anel, Anel, M.
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #math.CT
paper · pdf · doi:10.48550/arxiv.1409.4688
arxiv created 2014/09/16 · openalex publication_date 2014/09/16 · arxiv updated 2014/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a recursive formula to compute the cofree coalgebra P^\vee(C) over any colored operad P in Set, CGHaus or (dg)Vect. The construction is closed to that of Smith but different. We use a more conceptual approach to simplify the proofs that P^\vee is the cofree P-coalgebra functor and also the comonad generating P-coalgebras. In a second part, when P is a linear or dg-operad over a field, we generalize the notion of representative functions of Block & Leroux and prove that P^\vee(C) is simply the subobject of representative elements in the "completed P-algebra" P^\wedge(C). This says that our recursion (as well as that of Smith) stops at the first step.