Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parametrised functor calculus: excision, spheres, and semiadditivity

Kaif Hilman, Sil Linskens

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 Wirthmüller's classical result in equivariant homotopy theory for finite groups.

Parametrised functor calculus: excision, spheres, and semiadditivity

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 Wirthmüller's classical result in equivariant homotopy theory for finite groups.

Paper Structure

This paper contains 11 sections, 37 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Parametrised excision for posets
  3. Excisable posets
  4. Face decompositions
  5. Excision
  6. Parametrised cubical excision
  7. Cubes
  8. Suspensions and spheres
  9. Cubical excision
  10. Semiadditivity and stability
  11. Decomposition of colimit diagrams

Key Result

Theorem A

Let $\sigma\colon \breve{\underline{{L}}}\subseteq \underline{{L}}$ be an excisable poset. Let $\underline{{\mathcal{C}}},\underline{{\mathcal{D}}}\in\mathrm{Cat}_{{\mathcal{T}}}$ where $\underline{{\mathcal{C}}}$ has $\sigma$--left Kan extensions and a final object and $\underline{{\mathcal{D}}}$ i

Theorems & Definitions (101)

  • Definition 1.1: c.f. \ref{['defn:excisable_poset']}
  • Definition 1.2
  • Theorem A: Excisive approximations, c.f. \ref{['mainthm:excisive_approximation_left_adjoint']}
  • Theorem 1.3
  • Theorem B: c.f. \ref{['mainthm:spherically_faithful_implies_semiadditivity']}
  • Theorem C: c.f. \ref{['mainThm:sphere_invertible_equivalent_to_parametrised_stable']}
  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Remark 2.1.4
  • ...and 91 more