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 {\displaystyle R,U} together with five morphisms

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

Examples

Group objects

A group object is a special case of a groupoid object, where R = U {\displaystyle R=U} and s = t {\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.

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 → y ) = x , t ( x → y ) = y {\displaystyle s(x\to y)=x,\,t(x\to y)=y} , m ( f , g ) = g ∘ f {\displaystyle m(f,g)=g\circ f} , e ( x ) = 1 x {\displaystyle e(x)=1_{x}} and i ( f ) = f − 1 {\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).

Groupoid schemes

A groupoid S-scheme is a groupoid object in the category of schemes over some fixed base scheme S. If U = S {\displaystyle U=S} , then a groupoid scheme (where s = t {\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 R = U × G {\displaystyle 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 ⇉ t s U {\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 [ R ⇉ U ] {\displaystyle [R\rightrightarrows U]} be the category of ( R ⇉ U ) {\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

Notes

  1. ^ Algebraic stacks, Ch 3. § 1.
  2. ^ (Gillet 1984)

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