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

By Michael Ching. 2005

118 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d195ed163dd1e8cb57ca33fe611860046e3289fd

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In Algebra and Number Theory

1. Marco Maggesi
2. Victor Tourtchine
3. Mark Behrens
4. André Hirschowitz
5. G. Arone

In Mathematical Physics

1. Marco Maggesi
2. André Hirschowitz
3. G. Arone
4. Mark Behrens
5. Victor Tourtchine
6. Victor Turchin
7. Ryan Budney
8. Thomas Willwacher

Citations in the corpus

• The Goodwillie tower and the EHP sequence (2010, background): “], [AM99]. Ching observed that this description yields an equivalence @ i (Id) ’B(1;Comm;1)_ where B(1;Comm;1) i is the ith space of the operadic bar construction on the commutative operad in spectra [Chi05]. Using this description, Ching gives @ (Id) the structure of an operad, and therefore the la”
• Modules over monads and initial semantics (2010, methodology): “years ago by M. Markl ([16, 17]) and are commonly used by topologists (see e.g. [9, 14, 4 ]).”
• On the rational homology of high dimensional analogues of spaces of long knots (2011, background): “10 The existence of the map (10-3) is closely related to the fact that KF is the Koszul dual of a right –module F , and therefore is a right comodule over a suitable version of the Lie cooperad; see Fresse [13] and Ching [8] for more detail.”
• Modules over Monads and Linearity (2006, unknown): “[9,14,4]).”