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 $$\mathcal{V}$$ 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 $$V \to \mathcal{V}$$. It has an action by the affine line $$\mathbb{A}^1$$ corresponding to scalar multiplication.
 * A Picard stack is an example of a group-stack (or groupoid-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 that satisfy the typical compatibility conditions.
 * 1) a morphism $$\sigma: X \times G \to X$$,
 * 2) (associativity) a natural isomorphism $$\sigma \circ (m \times 1_X) \overset{\sim}\to \sigma \circ (1_X \times \sigma)$$, where m is the multiplication on G,
 * 3) (identity) a natural isomorphism $$1_X \overset{\sim}\to \sigma \circ (1_X \times e)$$, where $$e: S \to G$$ is the identity section of G,

If, more generally, G is a group-stack, one then extends the above using local presentations.