Groupoid object explained

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

R, U together with five morphismss, t: R \to U, e: U \to R, m: R x _{U,} R \to R, i: R \to R satisfying the following groupoid axiomss \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 x _{U,} R \to R are the two projections,

1. (associativity)

m \circ ( 1 _{R} x m) = m \circ (m x 1 _{R),}

1. (unit)

m \circ (e \circ s, 1 _{R)} = m \circ ( 1 _{R,} e \circ t) = 1 _{R,}

1. (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 .

Examples

Group objects

A group object is a special case of a groupoid object, where

R = U and 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 \to y) = x, t(x \to y) = y , m(f, g) = g \circ f , e(x) = 1 _{x} and 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 , 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, 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 x G , s the projection, t the given action. This determines a groupoid scheme.

Constructions

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.

See also

• Simplicial scheme