Groupoid object

In category theory, a branch of mathematics, a groupoid object in a category C admitting finite fiber products is 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.

Examples

Example: 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 ${\displaystyle i(f)=f^{-1}}$.

Incidentally, one can consider a notion of a semigroupoid (unital semigroup = a category with a single object); but, according to this example, that is nothing but a category; so a groupoid object is really a special case of a "category object", better known as a stack (or prestack).

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, for example in (Gillet 1984) harv error: no target: CITEREFGillet1984 (help), to convey the idea it is a generalization of algebraic groups and their actions. 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.

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.

Construction

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.

See also

• simplicial scheme

Notes

1. Algebraic stacks, Ch 3. § 1. harvnb error: no target: CITEREFAlgebraic_stacks (help)

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
• H. Gillet, Intersection theory on algebraic stacks and Q-varieties, J. Pure Appl. Algebra 34 (1984), 193–240, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983).