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 formula_1 together with five morphisms formula_2 satisfying the following groupoid axioms
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 formula_10, formula_11, formula_12 and formula_13.
Incidentally, one can consider a notion of a semigroupoid (unital semigroup = a category with a single object); 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 (or prestack).
A groupoid "S"-scheme is a groupoid object in the category of schemes over some fixed base scheme "S". If formula_14, then a groupoid scheme (where formula_15 are necessarily the structure map) 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 formula_16, "s" the projection, "t" the given action. This determines a groupoid scheme.
Construction.
Given a groupoid object ("R", "U"), the equalizer of formula_17, 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" (if any) 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 formula_18 be the category of formula_19-torsors. Then it is a category fibered in groupoids; in fact, (in a nice case), a Deligne–Mumford stack. Conversely, any DM stack is of this form.