Groupoid object


In category theory, a branch of mathematics, a groupoid object in a category C admitting finite fiber products is a pair of objects together with five morphisms satisfying the following groupoid axioms
  1. where the are the two projections,
  4. ,,.
A group object is a special case of a groupoid object.

Examples

Example: A groupoid object in the category of sets is precisely a groupoid in the usual sense: a category in which every morphism is an isomorphism. Indeed, given such a category C, take U to be the set of all objects in C, R the set of all arrows in C, the five morphisms given by,, and.
Incidentally, one can consider a notion of a semigroupoid ; but, according to this example, that is nothing but a category; so a groupoid object is really a special case of a "category object", better known as a stack.
A groupoid S-scheme is a groupoid object in the category of schemes over some fixed base scheme S. If, then a groupoid scheme is the same as a group scheme. A groupoid scheme is also called an algebraic groupoid, for example in, to convey the idea it is a generalization of algebraic groups and their actions. When the term "groupoid" can naturally refer to a groupoid object in some particular category in mind, the term groupoid set is used to refer to a groupoid object in the category of sets.
Example: Suppose an algebraic group G acts from the right on a scheme U. Then take, s the projection, t the given action. This determines a groupoid scheme.

Construction

Given a groupoid object, the equalizer of, if any, is a group object called the inertia group of the groupoid. The coequalizer of the same diagram, if any, is the quotient of the groupoid.
Each groupoid object in a category C may be thought of as a contravariant functor from C to the category of groupoids. This way, each groupoid object determines a prestack in groupoids. This prestack is not a stack but it can be stackified to yield a stack.
The main use of the notion is that it provides an atlas for a stack. More specifically, let be the category of -torsors. Then it is a category fibered in groupoids; in fact,, a Deligne–Mumford stack. Conversely, any DM stack is of this form.