Morphism of algebraic stacks


In algebraic geometry, given algebraic stacks over a base category C, a morphism of algebraic stacks is a functor such that.
More generally, one can also consider a morphism between prestacks; for this, see prestack#Morphisms

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 of relative dimension j for some smooth scheme U of dimension n. For example, if denotes the moduli stack of rank-n vector bundles, then there is a presentation given by the trivial bundle over.
A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.