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 $R,U$ together with five morphisms

$$\begin{gathered} s,t:R\to U, \\ e:U\to R, \\ m:R{\times }_{U,t,s}R\to R, \\ i:R\to R \end{gathered}$$

satisfying the following groupoid axioms

1. $s\circ e=t\circ e=1_U,s\circ m=s\circ p_1,t\circ m=t\circ p_2$ where the $p_i:R{\times }_{U,t,s}R\to R$ are the two projections,
2. (associativity) $m\circ (1_R\times m)=m\circ (m\times 1_R),$
3. (unit) $m\circ (e\circ s,1_R)=m\circ (1_R,e\circ t)=1_R,$
4. (inverse) $i\circ i=1_R$, $s\circ i=t,t\circ i=s$, $m\circ (1_R,i)=e\circ s,m\circ (i,1_R)=e\circ t$.

Examples

Group objects

A group object is a special case of a groupoid object, where $R=U$ and $s=t$. 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 arrows in C, the five morphisms given by $s(x\to y)=x,t(x\to y)=y$, $m(f,g)=g\circ f$, $e(x)=1_x$ and $i(f)=f^{-1}$. 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).

Groupoid schemes

A groupoid S-scheme is a groupoid object in the category of schemes over some fixed base scheme S. If $U=S$, then a groupoid scheme (where $s=t$ 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 $R=U\times G$, s the projection, t the given action. This determines a groupoid scheme.

Constructions

Given a groupoid object (R, U), the equalizer of $R\stackrel{s}{\underset{t}{\rightrightarrows }}U$, 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 $[R\rightrightarrows U]$ be the category of $(R\rightrightarrows U)$-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.

See also

• Simplicial scheme

References

• Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), Algebraic stacks, archived from the original on 2008-05-05, retrieved 2014-02-11
• Gillet, Henri (1984), "Intersection theory on algebraic stacks and Q-varieties", Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), Journal of Pure and Applied Algebra, 34 (2–3): 193–240, doi:10.1016/0022-4049(84)90036-7, MR 0772058