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 together with five morphisms satisfying the following groupoid axioms

  1. where the are the two projections,
  2. (associativity)
  3. (unit)
  4. (inverse) , , .[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 , , and .

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 , then a groupoid scheme (where 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 , s the projection, t the given action. This determines a groupoid scheme.

Construction

Given a groupoid object (R, U), the equalizer of , 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 be the category of -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.

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).


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.