Abstract: In a foundational series of papers Goodwillie laid the grounds for a theory of "Taylor Series" for homotopy theory, defining a tower of functors which interpolate between the stable and unstable homotopy type of a functor. Work of Heuts refined this picture by associating to an infinity category a tower of infinity categories which interpolate between the stabilization and the category in a compatible way. Key to the understanding of this tower is a sequence of natural transformations referred to as "Tate diagonals". I will describe some attempts to understand these Tate diagonals, and if time permits some relations between Heuts tower and the homotopy nilpotent groups of Biederman and Dwyer.