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 [math]\displaystyle{ R, U }[/math] together with five morphisms

[math]\displaystyle{ s, t: R \to U, \ e: U \to R, \ m: R \times_{U, t, s} R \to R, \ i: R \to R }[/math]

satisfying the following groupoid axioms

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

Examples

Group objects

A group object is a special case of a groupoid object, where [math]\displaystyle{ R = U }[/math] and [math]\displaystyle{ s = t }[/math]. 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 [math]\displaystyle{ s(x \to y) = x, \, t(x \to y) = y }[/math], [math]\displaystyle{ m(f, g) = g \circ f }[/math], [math]\displaystyle{ e(x) = 1_x }[/math] and [math]\displaystyle{ i(f) = f^{-1} }[/math]. 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 [math]\displaystyle{ U = S }[/math], then a groupoid scheme (where [math]\displaystyle{ s = t }[/math] 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 [math]\displaystyle{ R = U \times G }[/math], s the projection, t the given action. This determines a groupoid scheme.

Constructions

Given a groupoid object (R, U), the equalizer of [math]\displaystyle{ R \overset{s}\underset{t}\rightrightarrows U }[/math], 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 [math]\displaystyle{ [R \rightrightarrows U] }[/math] be the category of [math]\displaystyle{ (R \rightrightarrows U) }[/math]-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, http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1, retrieved 2014-02-11 
• Gillet, H. (1984), "Intersection theory on algebraic stacks and Q-varieties", Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), J. Pure Appl. Algebra, 34, pp. 193–240, http://ac.els-cdn.com/0022404984900367/1-s2.0-0022404984900367-main.pdf?_tid=a96230a2-515a-11e7-8661-00000aab0f27&acdnat=1497483728_c3a14f29c251aa4dc2b55b3abfa6ba9e