Parametrised functor calculus: excision, spheres, and semiadditivity

TL;DR

This paper develops a new parametrised functor calculus framework, introducing excisable posets and connecting excisiveness with semiadditivity, extending classical results in equivariant homotopy theory.

Contribution

It generalises parts of Goodwillie's functor calculus by defining excisable posets and relating excisiveness to semiadditivity in a parametrised setting.

Findings

Introduces the notion of excisable posets.

Establishes a link between excisiveness and semiadditivity.

Generalises Wirthmüller's classical equivariant homotopy result.

Abstract

We lay down the foundations of a theory of parametrised functor calculus, generalising parts of the functor calculus of Goodwillie. We introduce the notion of excisable posets and develop a theory of excisive approximations in this context. As an application, we introduce two different excisable posets when parametrising over an atomic orbital category. By comparing the notions of excisiveness for these two posets, we relate the invertibility of certain spheres with Nardin's notion of parametrised semiadditivity, generalising Wirthm\"uller's classical result in equivariant homotopy theory for finite groups.