Algebraic stack

In algebraic topology is meant by a stack ( English for " stack " ) a ( in a certain way ) kategorifizierte sheaf. The Kategorifizierung consists of two steps: the Kategorifizierung a presheaf and the descent axiom whose fulfillment makes a presheaf into a sheaf.

For a topological space is the category whose objects are surjective continuous maps, and whose morphisms are surjective continuous maps, such that.

• A presheaf on a category is a contravariant functor. For every morphism and a pullback

One gets an induced commuting diagram

( with inverted arrows ). According to the universal property of a pullback

There is a unique morphism in the category.

• The descent axiom for the presheaf is: For each of the morphism is an isomorphism.

It can be considered that these definitions coincide with the more common of the article on sheaves now. Each may permit Kategorifizierung in natural way: categories are 2- categories, functors are 2- functors, objects are categories, morphisms are functors, and equations of morphisms are natural equivalences. The category is a 2- category, by allowing only identities as 2- morphisms.

This results in the following definitions:

• A fibred category over in a 2- category is a contravariant 2- functor.
• The descent axiom for a fibred category is as follows: For each 1- morphism of the functor is an equivalence of categories.
• A stack is a fibered category which satisfies the descent axiom.

Remark: Actually should be a fibred category "Pre- stack" hot, but this term is already occupied by a different, non- equivalent definition.