Volume 10, issue 1 (2010)

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Bar constructions and Quillen homology of modules over operads

John E Harper

Algebraic & Geometric Topology 10 (2010) 87–136

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.

symmetric spectra, model category, operads, Quillen homology, chain complex
Mathematical Subject Classification 2000
Primary: 55P43, 55P48, 55U35, 18G55
Received: 14 February 2008
Revised: 27 September 2009
Accepted: 7 October 2009
Published: 10 January 2010
John E Harper
Institut de Géométrie, Algèbre et Topologie
CH-1015 Lausanne
Department of Mathematics
University of Notre Dame
Notre Dame, IN 46556