nLab Goodwillie-differentiable (infinity,1)-category

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 $\mathcal{C}$ is Goodwillie-differentiable if

1. it has finite (∞,1)-limits;

2. it has sequential (∞,1)-colimits;

3. the (∞,1)-colimit (∞,1)-functor $\underset{\longrightarrow}{\lim} Func(\mathbb{N}, \mathcal{C}) \longrightarrow \mathcal{C}$ 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 $(\infty,1)$-category.

(Lurie, example 7.1.1.8)

Properties

$n$-Excisive reflection / Taylor tower

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

Related concepts

• tangent (∞,1)-category

• jet (∞,1)-category

• Goodwillie calculus

References

• Jacob Lurie, Higher Algebra