Groupoid object


In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined.

Definition

A groupoid object in a category C admitting finite fiber products consists of a pair of objects together with five morphisms
satisfying the following groupoid axioms
  1. where the are the two projections,
  4. ,,.

Examples

Group objects

A group object is a special case of a groupoid object, where and. One recovers therefore topological groups by taking the category of topological spaces, or Lie groups by taking the category of manifolds, etc.

Groupoids

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 morphisms in C, the five morphisms given by,, and. 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.
However, unlike in the previous example with Lie groups, a groupoid object in the category of manifolds is not necessarily a Lie groupoid, since the maps s and t fail to satisfy further requirements.

Groupoid schemes

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, to convey the idea it is a generalization of algebraic groups and their actions.
For 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.

Constructions

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.