Goodwillie Calculus and Geometric Stacks

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 $\mathbb{R}F(A)\cong \text{holim}\mathbb{R}F(A_{\le n})$ for geometric stacks, where $\{A_{\le n}\}$ 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.

33 pages. The introduction has been rewritten in part to present this work from the perspective of the bridge technique of O. Caramello in a higher categorical setting, and corresponding additions have been made in the main results to that effect