# nLab groupoid object

## Idea

The notion of groupoid object is a horizontal categorification of a group object. Equivalently, it is an internal category which is a groupoid (in the appropriate internal sense.

## Definitions

Let $C$ be a category with finite limits, and let $\Grpd$ be the category of small groupoids. We can define groupoid objects representably:

###### Definition

A groupoid object in $C$ is a functor $F\colon C\to \Grpd$ such that

1. there is an object $g_0\in C_0$ such that there is a natural isomorphism $F(c)_0\simeq C(c,g_0)$

2. there is an object $g_1\in C_0$ such that there is a natural isomorphism $F(c)_1\simeq C(c,g_1)$

We can also define them more explicitly:

###### Definition

A groupoid object in $C$ is an internal category $A$ such that there is an “inverse-assigning morphism” $i\colon A_1 \to A_1$ satisfying certain axioms…

## Examples

• A groupoid in $\Top$ is a topological groupoid.

• A groupoid in $\Diff$ is a Lie groupoid. (Note that $\Diff$ does not have all pullbacks, but by suitable conditions on the source and target map we can ensure that the requisite pullbacks do exist.)

Revised on November 8, 2013 00:13:41 by Urs Schreiber (82.169.114.243)