Geometry & Topology, Vol. 9 (2005)
Paper no. 20, pages 833--934.
Bar constructions for topological operads and the Goodwillie derivatives of the identity
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
AMS subject classification.
Secondary: 18D50, 55P43.
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
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