Geometry & Topology

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

Michael Ching

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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.

Article information

Geom. Topol., Volume 9, Number 2 (2005), 833-934.

Received: 18 March 2005
Revised: 13 December 2005
Accepted: 6 May 2005
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P48: Loop space machines, operads [See also 18D50]
Secondary: 18D50: Operads [See also 55P48] 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)

operad cooperad bar construction module


Ching, Michael. Bar constructions for topological operads and the Goodwillie derivatives of the identity. Geom. Topol. 9 (2005), no. 2, 833--934. doi:10.2140/gt.2005.9.833.

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