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Functor calculus via non-cubes

Robin Stoll

TL;DR

The paper generalizes Goodwillie calculus to sigma-excisive functors indexed by preshapes and shapes beyond cubes, constructing universal excisive approximations $\operatorname{P}_{\sigma}$ via Pad and Lan and organizing them into the Taylor graph. It proves that, for full shapes with appropriate niceness/differentiability hypotheses, $\operatorname{P}_{\sigma}$ is left adjoint to inclusion and that the Taylor graph encodes all such approximations with maps induced by universal properties. A central finding is that the limits of the Taylor graph and the classical Taylor tower agree when restricted to non-inane shapes between finite posets, and the Taylor graph offers a potentially finer resolution of excision data when non-cubes are considered. The work develops a detailed shape-theoretic framework—distinguishing full, free, inane, and non-inane shapes, and analyzing indirect and direct maps between approximations—culminating in a program to relate general indexing shapes to the traditional cubic paradigm via cubical shapes and a conjectured all-encompassing cubes equivalence.

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.

Functor calculus via non-cubes

TL;DR

The paper generalizes Goodwillie calculus to sigma-excisive functors indexed by preshapes and shapes beyond cubes, constructing universal excisive approximations via Pad and Lan and organizing them into the Taylor graph. It proves that, for full shapes with appropriate niceness/differentiability hypotheses, is left adjoint to inclusion and that the Taylor graph encodes all such approximations with maps induced by universal properties. A central finding is that the limits of the Taylor graph and the classical Taylor tower agree when restricted to non-inane shapes between finite posets, and the Taylor graph offers a potentially finer resolution of excision data when non-cubes are considered. The work develops a detailed shape-theoretic framework—distinguishing full, free, inane, and non-inane shapes, and analyzing indirect and direct maps between approximations—culminating in a program to relate general indexing shapes to the traditional cubic paradigm via cubical shapes and a conjectured all-encompassing cubes equivalence.

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.
Paper Structure (30 sections, 110 theorems, 81 equations)

This paper contains 30 sections, 110 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Structure of this paper.
  3. Acknowledgments.
  4. Notation and conventions
  5. Excisive functors
  6. Preshapes
  7. Construction of the excisive approximation
  8. Shapes
  9. Proof of the excisive approximation
  10. Maps between approximations
  11. Indirect maps
  12. Direct maps
  13. The structure of shapes
  14. Full shapes
  15. Free shapes
  16. ...and 15 more sections

Key Result

Theorem A

Let $\sigma$ be a full shape and $F$ a functor. Then there is an explicit construction of its universal $\sigma$-excisive approximation $\mathop{\operatorname{P}_{\!\sigma}}F$.

Theorems & Definitions (289)

  • Definition
  • Theorem A
  • Theorem B
  • Theorem C
  • Conjecture
  • Remark 4
  • Remark 9
  • Remark 15
  • Definition 16
  • Remark 17
  • ...and 279 more