Use this tag for questions related to algebraic stacks, which are a stacks in groupoids X over the etale site such that the diagonal map of X is representable, and there exists a smooth surjection from a scheme to X.

An *algebraic stack* or *Artin stack* is a stack in groupoids $X$ over the etale site such that the diagonal map of $X$ is representable, and there exists a smooth surjection from (the stack associated to) a scheme to $X$. A morphism $Y \rightarrow X$ of stacks is *representable* if for every morphism $S \rightarrow X$ from (the stack associated to) a scheme to $X$, the fiber product $Y \times_X S$ is isomorphic to (the stack associated to) an algebraic space. The *fiber product* of stacks is defined using the usual universal property and changing the requirement that diagrams commute to the requirement that they 2-commute.

The motivation behind the representability of the diagonal is that the diagonal morphism $\Delta : \mathfrak X \to \mathfrak X \times \mathfrak X$ is representable if and only if for any pair of morphisms of algebraic spaces $X,$ $Y \to \mathfrak X$, their fiber product $X \times_{\mathfrak X} Y$ is representable.