# nLab differentiable (infinity,1)-category

Contents

### Context

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

(∞,1)-category theory

## Models

#### Stable Homotopy theory

stable homotopy theory

Introduction

# 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 $\mathcal{C}$ 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.

## Examples

###### Example

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

## Properties

### $n$-Excisive reflection / Taylor tower

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