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 $F:C\to Grpd$ 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” $i:A_1\to A_1$ 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.)

Related concepts

• monoid, monoid object, monoid object in an (∞,1)-category

• group, group object, group object in an (∞,1)-category

• groupoid, groupoid object, groupoid object in an (∞,1)-category

• infinity-groupoid, infinity-groupoid object, infinity-groupoid object in an (∞,1)-category?

• ring, ring object