Explicit construction of cofree precoalgebras and coalgebras

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.