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

Ching, Michael

doi:10.2140/gt.2005.9.833

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

Michael Ching

Geometry & Topology 9 (2005) 833–934

DOI: 10.2140/gt.2005.9.833

arXiv: math.AT/0501429

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

Mathematical Subject Classification 2000

Primary: 55P48

Secondary: 18D50, 55P43

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Publication

Received: 18 March 2005

Revised: 13 December 2005

Accepted: 6 May 2005

Published: 23 May 2005

Proposed: Thomas Goodwillie

Seconded: Ralph Cohen, Gunnar Carlsson

Authors

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

Volume 9, issue 2 (2005)