Definitions
A category c with a function to a category C is called a fibered category over C if it satisfies the following condition:
- For any morphism F from X to Y in C and any object y of c with image Y, there is a pullback f:x→y of y and X. This means that f is a universal morphism from some x to y with image F: any other morphism g:z→y with image F can be factored through f by a unique morphism from z to x. The element x=F*y is called the restriction of y to X and is unique up to canonical isomorphism.
The category c is called a prestack over a category C with a Grothendieck topology if it is fibered over C and in addition satisfies the following condition:
- For any object U of C and and objects x, y of c with image U, the functor from objects over U to sets taking F:V→U to Hom(F*x,F*y) is a sheaf.
The category c is called a stack over the category C with a Grothendieck topology if it is a prestack over C and in addition satisfies the following condition:
- Any decent datum is effective. A descent datum consists roughly of a covering of an object V of C by a family Vi, elements xi in the fiber over Vi, and morphisms fji between the restrictions of xi and xj to Vij=Vi×UVj satisfying the compatibility condition fki = fkjfji. The descent datum is called effective if the elements xi are essentially the restrictions of an element x with image U to the objects Vi.
A stack is called a stack in groupoids if it is also fibered in groupoids, meaning that its fibers (the inverse images of elements of C) are groupoids. Some authors use the word "stack" to refer to the more restrictive notion of a stack in groupoids.
Read more about this topic: Stack (descent Theory)
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