# 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 (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 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.[1]

## Notes

- § 8.6 of F. Meyer, Notes on algebraic stacks