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.
Categories: math.AT math.AG