Volume 9, issue 2 (2005)

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

Michael Ching

Geometry & Topology 9 (2005) 833–934

arXiv: math.AT/0501429


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.

operad, cooperad, bar construction, module
Mathematical Subject Classification 2000
Primary: 55P48
Secondary: 18D50, 55P43
Forward citations
Received: 18 March 2005
Revised: 13 December 2005
Accepted: 6 May 2005
Published: 23 May 2005
Proposed: Thomas Goodwillie
Seconded: Ralph Cohen, Gunnar Carlsson
Michael Ching
Department of Mathematics
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Massachusetts Institute of Technology
Massachusetts 02139