Ching, Michael Bar constructions for topological operads and the {G}oodwillie derivatives of the identity. (English) Zbl 1153.55006 Geom. Topol. 9, 833-933 (2005). The author introduces a general bar construction for operads in pointed symmetric monoidal categories which are enriched, tensored and cotensored over the category of based topological spaces. This bar construction applies to operads \(P\) such that \(P(0)\) is the zero object and \(P(1)\) is the unit of the given pointed symmetric monoidal category. The bar construction is defined as a coend by means of an interesting, new geometric realization of trees. The author shows that his bar construction comes equipped with a canonical cooperad structure, and is moreover isomorphic to the geometric realization of the standard simplicial bar construction. By dualization, he obtains a canonical operad structure on the Goodwillie derivatives, cf. T. G. Goodwillie [Geom. Topol. 7, 645–711 (2003; Zbl 1067.55006)] of the identity functor on based topological spaces, cf. G. Arone and M. Mahowald [Invent. Math. 135, 743–788 (1999; Zbl 0997.55016)].As opposed to the \(W\)-construction of J. M. Boardman and R. M. Vogt [Homotopy invariant algebraic structures on topological spaces. Lect. Notes in Math. 347 (1973; Zbl 0285.55012)] which (under suitable conditions) yields a cofibrant resolution of \(P\) in the category of operads, cf. C. Berger and I. Moerdijk [Topology 45, 807–849 (2006; Zbl 1105.18007)], the author’s bar construction is likely to define a cofibrant resolution of \(P\) in the category of \(P\)-bimodules, as is the case for the algebraic bar construction of V. Ginzburg and M. Kapranov [Duke Math. J. 76, 203–272 (1994; Zbl 0855.18006)]. In particular, the author relates the topological and algebraic bar constructions in an explicit way by means of a convergent homological spectral sequence. Reviewer: Clemens Berger (Nice) Cited in 3 ReviewsCited in 37 Documents MSC: 55P48 Loop space machines and operads in algebraic topology 18D50 Operads (MSC2010) 55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) Keywords:operad; cooperad; bar construction; module over an operad Citations:Zbl 1067.55006; Zbl 0997.55016; Zbl 0285.55012; Zbl 1105.18007; Zbl 0855.18006 PDFBibTeX XMLCite \textit{M. Ching}, Geom. Topol. 9, 833--933 (2005; Zbl 1153.55006) Full Text: DOI arXiv EuDML EMIS References: [1] G Arone, M Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999) 743 · Zbl 0997.55016 [2] C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. 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