nLab
differentiable (infinity,1)-category

Contents

Context

(,1)(\infty,1)-Category theory

Stable Homotopy theory

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. Note that this concept is not related to that of Lie ∞-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.

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 October 22, 2013 at 11:42:36. See the history of this page for a list of all contributions to it.