Functor calculus via non-cubes

Abstract

In this paper we study versions of Goodwillie's calculus of functors that use diagrams other than cubes. We first prove that the universal excisive approximation exists for a larger class of diagrams, which we call shapes. We then study the extension of the Taylor tower that contains all of these approximations. We prove that its limit agrees with the limit of the tower using certain criteria for the existence of maps between two excisive approximations. Lastly we investigate the question when a shape yields a notion of excision equivalent to one of the classical ones.