Sheaf on an algebraic stack

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack formula_1 is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is the data consists of, for each a scheme "S" in the base category and formula_2 in formula_3, a quasi-coherent sheaf formula_4 on "S" together with maps implementing the compatibility conditions among formula_4's.
For a Deligne–Mumford stack, there is a simpler description in terms of a presentation formula_6: a quasi-coherent sheaf on formula_1 is one obtained by descending a quasi-coherent sheaf on "U". A quasi-coherent sheaf on a Deligne–Mumford stack generalizes an orbibundle (in a sense).
Constructible sheaves (e.g., as ℓ-adic sheaves) can also be defined on an algebraic stack and they appear as coefficients of cohomology of a stack.
Definition.
The following definition is 
Let formula_1 be a category fibered in groupoids over the category of schemes of finite type over a field with the structure functor "p". Then a quasi-coherent sheaf is the data consisting of:
ℓ-adic formalism.
The ℓ-adic formalism (theory of ℓ-adic sheaves) extends to algebraic stacks.