The structure of simply colored coalgebras

Abstract

A simply colored coalgebra is a coassociative counital coalgebra C C over an arbitrary ring R R , which can be decomposed into a direct sum of two R 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 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.