## Algebraic & Geometric Topology

### Bar constructions and Quillen homology of modules over operads

John E Harper

#### 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.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882309

Digital Object Identifier
doi:10.2140/agt.2010.10.87

Mathematical Reviews number (MathSciNet)
MR2580430

Zentralblatt MATH identifier
1197.18002

#### Citation

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. https://projecteuclid.org/euclid.agt/1513882309

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