Groupoid object
A groupoid object in a category "C" admitting finite fiber products consists of a pair of objects formula_1 together with five morphisms
formula_2
satisfying the following groupoid axioms
# formula_3 where the formula_4 are the two projections,
# (associativity) formula_5
# (unit) formula_6
# (inverse) formula_7, formula_8, formula_9. A group object is a special case of a groupoid object, where formula_10 and formula_11. One recovers therefore topological groups by taking the category of topological spaces, or Lie groups by taking the category of manifolds, etc. 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_12, formula_13, formula_14 and formula_15. 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 (they are not necessarily submersions). A groupoid "S"-scheme is a groupoid object in the category of schemes over some fixed base scheme "S". If formula_16, then a groupoid scheme (where formula_11 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. For example, suppose an algebraic group "G" acts from the right on a scheme "U". Then take formula_18, "s" the projection, "t" the given action. This determines a groupoid scheme. Given a groupoid object ("R", "U"), the equalizer of formula_19, 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_20 be the category of formula_21-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.
Examples
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.