A generalized Blakers-Massey theorem and Goodwillie calculus

(joint with M. Anel, E. Finster, A. Joyal) We explain our generalized Blakers-Massey theorem. For this we introduce the notion of modality: a unique factorization system whose left class is closed under base change. The example of n-connected/n-truncated maps leads to the classical Blakers-Massey. In the context of Goodwillie calculus we find another example: factoring a natural transformation into a P_n-equivalence followed by an n-excisive map. This leads to a Blakers-Massey Theorem for the Goodwillie tower. This gives a quick and independent proof of the fact that homogeneous functors deloop.