The structure of simply colored coalgebras

TL;DR

This paper characterizes simply colored coalgebras over fields as pointed coalgebras with a specific splitting, and establishes their categorical completeness and cocompleteness.

Contribution

It introduces a new equivalence between simply colored coalgebras and pointed coalgebras with a splitting of the coradical.

Findings

Equivalence between simply colored coalgebras and pointed coalgebras with splitting

Category of simply colored coalgebras is complete and cocomplete

Provides structural insights into coalgebra decompositions

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.