nLab
Goodwillie-differentiable (infinity,1)-category

Contents

Contents

Idea

In the context of Goodwillie calculus, an (∞,1)-category is called Goodwillie-differentiable if (∞,1)-functors to it admit “derivatives” in the form of n-excisive approximations.

(This concept is not related to that of smooth ∞-groupoids.)

Definition

Definition

An (∞,1)-category 𝒞\mathcal{C} is Goodwillie-differentiable if

  1. it has finite (∞,1)-limits;

  2. it has sequential (∞,1)-colimits;

  3. the (∞,1)-colimit (∞,1)-functor limFunc(,𝒞)𝒞\underset{\longrightarrow}{\lim} Func(\mathbb{N}, \mathcal{C}) \longrightarrow \mathcal{C} is a left exact (∞,1)-functor, hence commutes with finite (∞,1)-limits.

(Lurie, def. 6.1.1.6)

Examples

Example

Every (∞,1)-topos is a Goodwillie-differentiable (,1)(\infty,1)-category.

(Lurie, example 7.1.1.8)

Properties

nn-Excisive reflection / Taylor tower

By Goodwillie calculus, (∞,1)-functors to Goodwillie-differentiable (,1)(\infty,1)-categories have n-excisive approximations/Taylor towers.

References

Last revised on January 5, 2016 at 05:17:35. See the history of this page for a list of all contributions to it.