Bar constructions and Quillen homology of modules over operads

Harper, John E

doi:10.2140/agt.2010.10.87

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Bar constructions and Quillen homology of modules over operads

John E Harper

Algebraic & Geometric Topology 10 (2010) 87–136

DOI: 10.2140/agt.2010.10.87

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.

Keywords

symmetric spectra, model category, operads, Quillen homology, chain complex

Mathematical Subject Classification 2000

Primary: 55P43, 55P48, 55U35, 18G55

References

Publication

Received: 14 February 2008

Revised: 27 September 2009

Accepted: 7 October 2009

Published: 10 January 2010

Authors

John E Harper
Institut de Géométrie, Algèbre et Topologie EPFL CH-1015 Lausanne Switzerland	Department of Mathematics University of Notre Dame Notre Dame, IN 46556 USA

Volume 10, issue 1 (2010)