Holmstrom Quillen homology

Quillen homology

http://hopf.math.purdue.edu/cgi-bin/generate?/Harper/QuillenHomology

Title: Bar constructions and Quillen homology of modules over operads Author: John E. Harper Author’s mailing address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA

Comments: 33 pages, uses xy-pic; compiled the .tex file without using the dvips,ps options in xy-pic, to ensure .dvi is device independent, but diagrams may now appear jagged, etc.

Abstract: This paper shows that Quillen derived homology of modules and algebras over an operad, for symmetric sequences of symmetric spectra and unbounded chain complexes, can be calculated using simplicial bar constructions, modulo cofibrancy conditions. Working with several model category structures, a homotopical proof is given, after showing that certain homotopy colimits in modules and algebras over an operad can be easily understood. The key result here, which is at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits.


See Goerss-Schemmerhorn p. 30 for a nice discussion. Includes abelianization in model categories, and the Quillen homology of an object as the total left derived functor of abelianization. Can recover singular homology as a special case. Also group homology (degree-shifted) and Andre-Quillen homology of commutative algebras.

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Quillen homology

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Quillen homology

arXiv:1102.1234 On Quillen homology and a homotopy completion tower for algebras over operads from arXiv Front: math.AT by John E. Harper, Kathryn Hess We describe and study a (homotopy) completion tower for algebras and left modules over operads in symmetric spectra. We prove that a weak equivalence on topological Quillen homology induces a weak equivalence on homotopy completion, and that for 00-connected algebras and modules over a 1-1-connected operad, the homotopy completion tower interpolates between topological Quillen homology and the identity functor. By an explicit calculation of its layers, we show that the homotopy completion tower is the precise analog—in the context of algebras and modules over operads—of the Goodwillie tower of the identity functor

As easy consequences of the strong convergence properties of the homotopy completion tower, we prove a Whitehead theorem and a Hurewicz theorem for topological Quillen homology. We also prove a relative Hurewicz theorem that provides conditions under which topological Quillen homology detects nn-connected maps. We prove a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups; this result should be thought of as an algebras over operads in spectra analog of Serre’s finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to topological Quillen homology, and use this to define Quillen homology completion—in the sense of Bousfield-Kan—for algebras and modules over operads. We also prove analogous results for algebras and modules over operads in unbounded chain complexes.

nLab page on Quillen homology

Created on June 10, 2014 at 21:14:54 by Andreas Holmström