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.

A groupoid object in a category C admitting finite fiber products consists of a pair of objects ${\displaystyle R,U}$ together with five morphisms

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

satisfying the following groupoid axioms

1. ${\displaystyle s\circ e=t\circ e=1_{U},\,s\circ m=s\circ p_{1},t\circ m=t\circ p_{2}}$ where the ${\displaystyle p_{i}:R\times _{U,t,s}R\to R}$ are the two projections,
2. (associativity) ${\displaystyle m\circ (1_{R}\times m)=m\circ (m\times 1_{R}),}$
3. (unit) ${\displaystyle m\circ (e\circ s,1_{R})=m\circ (1_{R},e\circ t)=1_{R},}$
4. (inverse) ${\displaystyle i\circ i=1_{R}}$, ${\displaystyle s\circ i=t,\,t\circ i=s}$, ${\displaystyle m\circ (1_{R},i)=e\circ s,\,m\circ (i,1_{R})=e\circ t}$.^[1]

A group object is a special case of a groupoid object, where ${\displaystyle R=U}$ and ${\displaystyle s=t}$. 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 ${\displaystyle s(x\to y)=x,\,t(x\to y)=y}$, ${\displaystyle m(f,g)=g\circ f}$, ${\displaystyle e(x)=1_{x}}$ and ${\displaystyle i(f)=f^{-1}}$. 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).

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

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

• Simplicial scheme

• 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