Bar constructions and Quillen homology of modules over operads

TL;DR

This paper demonstrates that topological Quillen homology of operad modules can be computed via bar constructions, providing homotopical proofs and extending results to chain complexes.

Contribution

It introduces a homotopical approach to calculating Quillen homology using bar constructions across different model categories.

Findings

Bar constructions effectively compute Quillen homology in symmetric spectra.

The forgetful functor commutes with certain homotopy colimits.

Results extend to unbounded chain complexes.

Abstract

We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical proof after showing that certain homotopy colimits in algebras and modules over operads can be easily understood. A key result here, which lies at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. We also prove analogous results for algebras and modules over operads in unbounded chain complexes.