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Wednesday, March 18, 2015

**Abstract:** The goal of this talk is to understand the comonad that acts on the derivatives of a functor from based spaces to spectra. On the one hand, this comonad captures the structure of a `divided power' module over the Lie operad. I will give a different description in terms of the Koszul duals of the E_n-operads. In particular, the partially-stabilized cross-effects of such a functor have an action by K(E_n). Taking the colimit we get a comonad that acts on the derivatives themselves. Underlying this construction is a theory of Koszul duality for modules over operads of spectra. This is joint work with Greg Arone.