Grothendieck topologies, fibered categories and descent theory: an introduction to the stack theory (2015)
The notion of stack grew out of attempts to parameterize geometric objects varying in families. Such families are classically know as Moduli and their understanding is a central theme in Algebraic Geometry. The formalism of stacks was introduced by A. Grothendieck and M. Artin as the natural context in which moduli problems of objects with symmetries can be tackled. The usual spaces used in geometry– Manifolds, Varieties, Schemes– are inadequate for the parametrization of geometry objects that are self similar. Instead one needs a procedure that will not only encode the way things vary in families but will also remembers the intrinsic symmetries of each object in the family. As an example, let us consider two families of ellipses parametrized by a circle: The left-hand family is trivial, all the ellipses in this one are positioned in exactly the same way. The ellipses in the right-hand family are all the same but they are rotated in their planes. Families of this kind are called isotrivial. The usual parameter spaces do not capture the distinction between trivial and isotrivial families.