Goodwillie Calculus and Geometric Stacks

TL;DR

This paper explores the parallels between Goodwillie calculus of functors and the theory of geometric stacks, focusing on convergence, tower approximations, and homotopy limits in a homotopical algebraic context.

Contribution

It establishes a conceptual link between Goodwillie calculus and geometric stacks, demonstrating similar properties and convergence behaviors in their respective tower constructions.

Findings

Homotopy limits of functors relate to Postnikov towers of stacks

Parallel structures in homotopy fibers of towers

Reconstruction theorems for polynomial approximations

Abstract

We show Goodwillie's calculus of functors and $n$-geometric $D^{-}$-stacks share similar features by starting to focus on the convergence of Taylor towers for homotopy functors and the fact that $\mathbb{R} F(A) \cong \text{holim} \mathbb{R} F(A_{\leq n})$ for geometric stacks, where $\{ A_{\leq n} \}$ provides a Postnikov tower of some given $A \in s \text{k-Alg}$. From there we show parallel results, such as similar homotopy fibers of connecting maps in towers, as well as polynomial approximations, pointwise approximations and reconstruction theorems for towers.