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

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. Further, we 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. Finally, we extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that based spaces naturally give rise to modules over the operad formed by the derivatives of the identity.