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 $R,U$ together with five morphisms $s,t:R\to U,e:U\to R,m:R\times _{U,t,s}R\to R,i:R\to R$ satisfying the following groupoid axioms

$s\circ e=t\circ e=1_{U},\,s\circ m=s\circ p_{1},t\circ m=t\circ p_{2}$ where the $p_{i}:R\times _{U,t,s}R\to R$ are the two projections,
(associativity) $m\circ (1_{R}\times m)=m\circ (m\times 1_{R}),$
(unit) $m\circ (e\circ s,1_{R})=m\circ (1_{R},e\circ t)=1_{R},$
(inverse) $i\circ i=1_{R}$ , $s\circ i=t,\,t\circ i=s$ , $m\circ (1_{R},i)=e\circ s,\,m\circ (i,1_{R})=e\circ t$ .[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 $s(x\to y)=x,\,t(x\to y)=y$ , $m(f,g)=g\circ f$ , $e(x)=1_{x}$ and $i(f)=f^{-1}$ .

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 $U=S$ , 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, for example in (Gillet 1984), 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 $R=U\times G$ , s the projection, t the given action. This determines a groupoid scheme.

Construction

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

Notes

Algebraic stacks, Ch 3. § 1.

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).

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[1] Algebraic stacks, Ch 3. § 1.

Groupoid object

Examples

Construction

See also

Notes

References