Group stack

In algebraic geometry, a group stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way. It generalizes a group scheme, which is a scheme whose sets of points have group structures in a compatible way.

Examples

A group scheme is a group-\ stack. More generally, a group algebraic-space, an algebraic-space analog of a group scheme, is a group-stack.
Over a field k, a vector bundle stack on a Deligne–Mumford stack X is a group-stack such that there is a vector bundle V over k on X and a presentation. It has an action by the affine line corresponding to scalar multiplication.
A Picard stack is an example of a group-stack.

Actions of group stacks

The definition of a group action of a group stack is a bit tricky. First, given an algebraic stack X and a group scheme G on a base scheme S, a right action of G on X consists of

a morphism,
a natural isomorphism, where m is the multiplication on G,
a natural isomorphism, where is the identity section of G,

that satisfy the typical compatibility conditions.
If, more generally, G is a group stack, one then extends the above using local presentations.