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

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
  1. a morphism,
  2. a natural isomorphism, where m is the multiplication on G,
  3. 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.