Functor calculus via non-cubes

TL;DR

This paper extends Goodwillie's calculus of functors to more general diagram types, constructing universal excisive approximations and analyzing their properties to broaden the applicability of the Taylor tower framework.

Contribution

It introduces a generalized approach to functor calculus for non-cube diagrams, expanding the classical Taylor tower with new excisive approximations and establishing their relationships.

Findings

Constructed universal excisive approximations for broader diagram classes

Proved the equivalence of limits between extended and classical Taylor towers

Analyzed conditions under which new excision notions match classical ones

Abstract

We study versions of Goodwillie's calculus of functors for indexing diagrams other than cubes. We in particular construct universal excisive approximations for a larger class of diagrams, which yields an extension of the Taylor tower. We prove that the limit of this extension agrees with the limit of the Taylor tower using criteria for the existence of maps between excisive approximations. Lastly we investigate in which cases our new notions of excision coincide with classical ones.