# nLab equivalence of (2,1)-categories

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

The concept of equivalence of (2,1)-categories is the concept of equivalence of (infinity,1)-categories restricted along the inclusion of (2,1)-categories into all (infinity,1)-categories.

Equivalently, the concept of equivalence of 2-categories restricted along the inclusion of $(2,1)$-categories into 2-categories.

## Properties

Hence in the presence of the axiom of choice, a (2,1)-functor $f \colon \mathcal{C} \longrightarrow \mathcal{D}$ is an equivalence precisely if

1. it is essentially surjective, hence surjective on equivalence classes of objects;

2. it is fully faithful in that for all objects $c_1,c_2\in \mathcal{C}$ the 1-functor on hom-groupoids $f_{c_1,c_2}\colon \mathcal{C}(c_1,c_2) \longrightarrow \mathcal{D}(f(c_1),f(c_2))$ is an equivalence of groupoids.

Created on April 16, 2015 at 07:17:57. See the history of this page for a list of all contributions to it.