Morphism of algebraic stacks

In algebraic geometry, given algebraic stacks $p:X\to C,\,q:Y\to C$ over a base category C, a morphism $f:X\to Y$ of algebraic stacks is a functor such that $q\circ f=p$.

More generally, one can also consider a morphism between prestacks (a stackification would be an example).

Types

One particular important example is a presentation of a stack, which is widely used in the study of stacks.

An algebraic stack X is said to be smooth of dimension n - j if there is a smooth presentation $U\to X$ of relative dimension j for some smooth scheme U of dimension n. For example, if $\operatorname {Vect} _{n}$ denotes the moduli stack of rank-n vector bundles, then there is a presentation $\operatorname {Spec} (k)\to \operatorname {Vect} _{n}$ given by the trivial bundle $\mathbb {A} _{k}^{n}$ over $\operatorname {Spec} (k)$.

A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.

References

• Stacks Project, Ch, 83, Morphisms of algebraic stacks