An algebraic geometry stack is a presheaf of categories, with coherent restriction functors, that satisfies descent for both morphisms and objects. Informally, it is a sheaf of categories. More precisely, morphisms between two objects form a sheaf, and objects equipped with compatible local isomorphisms can be glued. This is the description by effective descent (Brylinski 1993, Stacks Project, Definition 8.4.1).
For example, consider the assignment to an open set of its category of sheaves of groups. The following properties express descent in this example:
1. For two sheaves 
and 
on a topological space 
,
the assignment 
is a sheaf, denoted 
.
2. Let 
be a surjective local homeomorphism
onto an open set 
. A sheaf 
on 
with an isomorphism 
on the fiber
product 
descends to a sheaf 
on 
provided
 |
on 
.
Here, 
and 
are the indicated projections. There is an isomorphism 
compatible with 
, and the pair 
is unique up to a unique
compatible isomorphism.
