Groupoid object

In , 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. A groupoid object in a C admitting finite s consists of a pair of together with five morphisms satisfying the following groupoid axioms where the are the two projections, (associativity) (unit) (inverse) , , . A group object is a special case of a groupoid object, where and . One recovers therefore topological groups by taking the , or Lie groups by taking the , etc. A groupoid object in the 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 . 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 , then a groupoid scheme (where are necessarily the structure map) 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. Given a groupoid object (R, U), 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 (if any) may be thought of as a contravariant functor from C to the category of groupoids.