Geometry & Topology, Vol. 9 (2005) Paper no. 20, pages 833--934.

Bar constructions for topological operads and the Goodwillie derivatives of the identity

Michael Ching

Abstract. We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.

Keywords. Operad, cooperad, bar construction, module

AMS subject classification. Primary: 55P48. Secondary: 18D50, 55P43.

DOI: 10.2140/gt.2005.9.833

E-print: arXiv:math.AT/0501429

Submitted to GT on 18 March 2005. Paper accepted 6 May 2005. Paper published 23 May 2005. Revised 13 December 2005 (reference to Salvatore added).

Notes on file formats

Michael Ching
Department of Mathematics, Room 2-089
Massachusetts Institute of Technology
Cambridge, MA 02139, USA

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