The structure of simply colored coalgebras

Abstract

A simply colored coalgebra is a coassociative counital coalgebra $C$ over an arbitrary ring $R$, which can be decomposed into a direct sum of two $R$-modules: one generated by set-like elements and another consisting of conilpotent elements. Our main result is the equivalence between simply colored coalgebras over a field $k$ and pointed coalgebras with a choice of splitting of its coradical. Additionally, we also prove that the category of simply colored coalgebras is both complete and cocomplete.

rewrite the abstract and introduction. Also, clarify some definitions and results based on advice from referees. To appear in Contemporary Math volume Higher Structures in Topology, Geometry and Physics. All comments are welcome