On Sweedler's cofree cocommutative coalgebra

TL;DR

This paper provides a direct proof of Sweedler's description of the cofree cocommutative coalgebra over a vector space, using local cohomology and residues to construct explicit liftings of maps into this universal coalgebra.

Contribution

It offers a new, explicit construction of the cofree cocommutative coalgebra and a direct proof of Sweedler's result utilizing local cohomology techniques.

Findings

Explicit construction of liftings into the cofree coalgebra

A direct proof of Sweedler's description

Application of local cohomology and residues

Abstract

We give a direct proof of a result of Sweedler describing the cofree cocommutative coalgebra over a vector space, and use our approach to give an explicit construction of liftings of maps into this universal coalgebra. The basic ingredients in our approach are local cohomology and residues.