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Operads and Chain Rules for the Calculus of Functors
"... Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain ..."
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Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this. In a landmark series of papers, [16], [17] and [18], Goodwillie outlines his ‘calculus of homotopy functors’. Let F: C → D (where C and D are each either Top ∗, the category of pointed topological spaces, or Spec, the category of spectra) be a pointed homotopy functor. One of the things that Goodwillie does is associate with F a sequence of spectra, which are called the derivatives of F.
Homotopy theory of modules over operads and nonΣ operads in monoidal model categories
 J. Pure Appl. Algebra
"... There are many interesting situations in which algebraic structure can be described ..."
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There are many interesting situations in which algebraic structure can be described
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY OF MODULES OVER OPERADS
, 802
"... There are many situations in algebraic topology, homotopy theory, and homological algebra in which operads parametrize interesting algebraic structures [10, 16, 27, 30, 35]. In many of these, there is a notion of abelianization or stabilization which provides a notion of homology [1, 2, 14, 42, 44]. ..."
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There are many situations in algebraic topology, homotopy theory, and homological algebra in which operads parametrize interesting algebraic structures [10, 16, 27, 30, 35]. In many of these, there is a notion of abelianization or stabilization which provides a notion of homology [1, 2, 14, 42, 44]. In these contexts,