# nLab groupoid object

Contents

## Internal $n$-category

#### Categorical algebra

internalization and categorical algebra

universal algebra

categorical semantics

# Contents

## Idea

A groupoid object in an ambient category $\mathcal{C}$ is equivalently

• a groupoid internal to $\mathcal{C}$;

• an internal category in $\mathcal{C}$ equipped with an inverse-assigning morphism.

This notion is the horizontal categorification of that of a group object.

## 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.)

The general definition of internal categories seems to have first been formulated in:

following the general principle of internalization formulated in

The concept of topological groupoids and Lie groupoids goes back to

• Charles Ehresmann, Catégories topologiques et categories différentiables, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 (pdf, zbMath:0205.28202)

and their understanding as categories internal to TopologicalSpaces and to SmoothManifolds is often attributed to

but it seems that the definition is not actually contained in there, certainly not in its simple and widely understood form due to Grothendieck 61.

Discussion of cartesian closed 2-categories of internal groupoids (mostly in Top, hence for topological groupoids):