Morphism of algebraic stacks

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

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

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

A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.¹