Categorical Foundations for Goodwillie's calculus of homotopy functors

Georg Biedermann
Thursday, July 25, 2013 - from 17:00 to 17:25
Tom Goodwillie introduced in the 80s and 90s a framework to study homotopy functors in a similar spirit to studying analytic functions by their Taylor expansion. A homotopy functor preserves weak equivalences and, originally, mapped from pointed or unpointed spaces or spectra to any of these categories. He defined a tower of natural transformations $$ \hdots\to P_nF\to P_{n-1}F\to\hdots\to P_1F\to P_0F $$ under $F$, where $P_nF$ is the closest approximation to $F$ among polynomial functors of degree $n$. Here, the functor $\pi_*P_1F$ is a generalized homology theory in the sense of Eilenberg and Steenrod. He also gave a classification of the $n$-homogeneous layers $D_nF=\hofib[P_nF\to P_{n-1}F]$ in terms of symmetric multilinear functors and spectra with $\Sigma_n$-action. I will first give an introduction to ``calculus''. If time permits I will mention the relation of the Goodwillie tower of the identity functor of pointed spaces with lower central series of Kan's loop group functor. Then I will talk about recent joint work with Oliver R\"ondigs (building on earlier joint work with Boris Chorny) describing the polynomial stages $P_n$, the homogeneous layers $D_n$, the cross effects and the differentials in terms of model structures and Quillen equivalences and generalizing the setup to functors between simplicial model cateogries satisfying some reasonable technical assumptions.