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.
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 A group object is a special case of a groupoid object, where
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.
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
are necessarily the structure map) is the same as a group scheme.
A groupoid scheme is also called an algebraic groupoid,[2] 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.
The coequalizer of the same diagram, if any, is the quotient of the groupoid.
The main use of the notion is that it provides an atlas for a stack.
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.