Lectures on bar and cobar

TL;DR

This paper explores the classical and derived bar and cobar constructions within an abstract framework for infinity-operads, providing new proofs of adjunctions and recovering classical comparison maps.

Contribution

It introduces a general abstract framework for bar and cobar constructions applicable to infinity-operads, offering new conceptual proofs and connections to classical results.

Findings

Provides new existence proofs for Lurie's adjunctions.

Recovers classical comparison maps like Szczarba and Hess-Tonks.

Unifies classical and derived bar/cobar constructions in a broad setting.

Abstract

We discuss Lurie's (derived) bar and cobar constructions, the classical ones for simplicial groups and sets (due to Eilenberg-MacLane and Kan), and the classical ones for differential graded (co)algebras (due to Eilenberg-MacLane and Adams) and their relations, putting them into an abstract framework which makes sense much more generally for any cofibration of infinity-operads. Along these lines we give new and rather conceptual existence proofs of Lurie's adjunction (where bar is left adjoint) and the classical adjunction (where bar is right adjoint). We also recover various classical comparison maps, e.g. the Szczarba and Hess-Tonks maps comparing Adams cobar with Kan's loop group.