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A Context for Manifold Calculus

Kensuke Arakawa

TL;DR

This work extends Weiss's manifold calculus to ∞-categories with small limits, introducing $k$-excisive (polynomial) and good functors and constructing universal polynomial approximations $T_kF$ via left Kan extensions. It provides a full classification of homogeneous degree-$k$ functors in terms of configuration spaces $B_k(M)$ and the functor $Sing\,B_k(M)$ when the target is pointed, recovering the classical Weiss classification in spaces and generalizing to arbitrary targets such as spectra or chain complexes. The Taylor tower framework is developed through Taylor cotowers, with convergence criteria tied to excisivity, exhaustivity, and analyticity, including connectivity results in the presence of a t-structure. The paper then extends these ideas to context-free manifold calculus and to the boundary case, showing that the same structural picture—polynomial approximations, tower convergence, and homogeneous classification—persists under appropriate replacements of disks, opens, and fat diagonals. Collectively, the results provide a flexible, ∞-categorical foundation for manifold calculus applicable to diverse targets and to invariants arising in factorization homology and related theories.

Abstract

We develop Weiss's manifold calculus in the setting of $\infty$-categories, where we allow the target $\infty$-category to be any $\infty$-category with small limits. We will establish the connection between polynomial functors, Kan extensions, and Weiss sheaves, and will classify homogeneous functors. We will also generalize Weiss and Boavida de Brito's theorem to functors taking values in arbitrary $\infty$-categories with small limits.

A Context for Manifold Calculus

TL;DR

This work extends Weiss's manifold calculus to ∞-categories with small limits, introducing -excisive (polynomial) and good functors and constructing universal polynomial approximations via left Kan extensions. It provides a full classification of homogeneous degree- functors in terms of configuration spaces and the functor when the target is pointed, recovering the classical Weiss classification in spaces and generalizing to arbitrary targets such as spectra or chain complexes. The Taylor tower framework is developed through Taylor cotowers, with convergence criteria tied to excisivity, exhaustivity, and analyticity, including connectivity results in the presence of a t-structure. The paper then extends these ideas to context-free manifold calculus and to the boundary case, showing that the same structural picture—polynomial approximations, tower convergence, and homogeneous classification—persists under appropriate replacements of disks, opens, and fat diagonals. Collectively, the results provide a flexible, ∞-categorical foundation for manifold calculus applicable to diverse targets and to invariants arising in factorization homology and related theories.

Abstract

We develop Weiss's manifold calculus in the setting of -categories, where we allow the target -category to be any -category with small limits. We will establish the connection between polynomial functors, Kan extensions, and Weiss sheaves, and will classify homogeneous functors. We will also generalize Weiss and Boavida de Brito's theorem to functors taking values in arbitrary -categories with small limits.
Paper Structure (17 sections, 54 theorems, 143 equations, 2 figures)

This paper contains 17 sections, 54 theorems, 143 equations, 2 figures.

Table of Contents

  1. Polynomial Functors
  2. Taylor Towers and Their Convergence
  3. Classification of Homogeneous Functors
  4. Context-Free Case
  5. Boundary Case
  6. (Non)-Localization Theorems
  7. Ayala--Francis's Theorem on Localizations at Isotopy Equivalences
  8. Global case
  9. A Remark on Classifying Spaces of Groupoids Internal to Top
  10. Some Results on $\infty$-Categories
  11. Mapping Spaces of Arrow Categories
  12. Kan Extension
  13. t-Structure and Homotopy Groups
  14. Extension by Zero Objects
  15. Miscellany
  16. ...and 2 more sections

Key Result

Theorem 7

Weiss99 Let $G:\operatorname{Disj}_{{\rm sm}}^{\leq k}\left(M\right)^{\mathrm{op}}\to\sf{Spaces}$ be a functor carrying isotopy equivalences to homotopy equivalences. The homotopy right Kan extension of $G$ along the inclusion $\operatorname{Disj}_{{\rm sm}}^{\leq k}\left(M\right)^{\mathrm{op}}\hook

Figures (2)

  • Figure 1.1: Picture of Lemma \ref{['lem:removing_disks']}.
  • Figure 1.2: Graph of $\phi$.

Theorems & Definitions (151)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 7
  • Theorem 8
  • Theorem 9: Theorem \ref{['thm:best_approx_exists']}
  • Theorem 10: Corollary \ref{['cor:homog_class']}, Theorem \ref{['thm:inv']}
  • Remark 11
  • Remark 12
  • Remark 13
  • ...and 141 more