  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.

Algebraic Topology;Category Theory