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

Definition

A groupoid object in CC is a functor F:CGrpdF\colon C\to \Grpd such that

  1. there is an object g 0C 0g_0\in C_0 such that there is a natural isomorphism F(c) 0C(c,g 0)F(c)_0\simeq C(c,g_0)

  2. there is an object g 1C 0g_1\in C_0 such that there is a natural isomorphism F(c) 1C(c,g 1)F(c)_1\simeq C(c,g_1)

We can also define them more explicitly:

Definition

A groupoid object in CC is an internal category AA such that there is an “inverse-assigning morphism” i:A 1A 1i\colon A_1 \to A_1 satisfying certain axioms…

Examples

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

  • A groupoid in Diff\Diff is a Lie groupoid. (Note that Diff\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)