Cellular coalgebras over the Barratt-Eccles operad I

TL;DR

This paper introduces a class of cellular coalgebras over the Barratt-Eccles operad that classify Z-completions of pointed, reduced simplicial sets, generalizing Quillen's rational homotopy theory to nilpotent cases.

Contribution

It establishes a new correspondence between cellular coalgebras over the Barratt-Eccles operad and homotopy types of nilpotent simplicial sets, extending classical rational homotopy results.

Findings

Classifies Z-completions of pointed, reduced simplicial sets

Encapsulates homotopy types of nilpotent simplicial sets

Generalizes Quillen's rational homotopy classification

Abstract

This paper considers a class of coalgebras over the Barratt-Eccles operad and shows that they classify Z-completions of pointed, reduced simplicial sets. As a consequence, they encapsulate the homotopy types of nilpotent simplicial sets. This result is a direct generalization of Quillen's result characterizing rational homotopy types via cocommutative coalgebras.