Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads

TL;DR

This paper explores operadic bar and cobar constructions to build cofibrant replacements and cylinder objects, enabling a better understanding of homotopy morphisms of operad algebras in differential graded contexts.

Contribution

It provides explicit constructions of cofibrant replacements and cylinder objects for operads, and demonstrates their use in characterizing homotopy morphisms as left homotopies.

Findings

Bar duality constructions work in unbounded dg modules

Explicit cylinder objects for operads are constructed

Homotopy morphisms are shown to be equivalent to left homotopies

Abstract

The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that the constructions of the bar duality work properly for algebras over operads in unbounded differential graded modules over a ring. In a second part, we use the operadic cobar construction to define explicit cyclinder objects in the category of operads. Then we apply this construction to prove that certain homotopy morphisms of algebras over operads are equivalent to left homotopies in the model category of operads.