Algebraic & Geometric Topology

Bar constructions and Quillen homology of modules over operads

John E Harper

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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.

Article information

Algebr. Geom. Topol., Volume 10, Number 1 (2010), 87-136.

Received: 14 February 2008
Revised: 27 September 2009
Accepted: 7 October 2009
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55P48: Loop space machines, operads [See also 18D50] 55U35: Abstract and axiomatic homotopy theory 18G55: Homotopical algebra

symmetric spectra model category operads Quillen homology chain complex


Harper, John E. Bar constructions and Quillen homology of modules over operads. Algebr. Geom. Topol. 10 (2010), no. 1, 87--136. doi:10.2140/agt.2010.10.87.

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