Explicit construction of cofree precoalgebras and coalgebras

TL;DR

This paper presents an explicit method for constructing cofree coalgebras and precoalgebras without restrictions on the coefficient ring, using infinite trees to generate homomorphisms.

Contribution

It introduces a novel explicit construction of cofree precoalgebras and coalgebras that does not require the coefficient ring to be a field or have other restrictions.

Findings

Constructs cofree coalgebras explicitly using infinite trees.

Provides a general method applicable over any coefficient ring.

Avoids the need for coassociativity or counital conditions in initial steps.

Abstract

In this paper, we will explicitly construct cofree coalgebras, by first constructing cofree precoalgebras (namely those not necessarily coassociative or counital). Our approach does not impose any condition to the coefficient ring, which means, for example, it does not need to be a field. The main technique is to generate homomorphisms from certain type of infinite trees so that those constitute cofree precoalgebras.