Stacks and Gerbes

A basic structure in mathematics for the study of a space X is to give to each open set an object A(U) in a category $\mathcal{c}$ together with restriction morphisms $r_{V, U}:A(U)\to A(V)$ for $V\subset U$ satisfying $r_{U,U}$ is the identity and the composition property

$$r_{W,U}=r_{W,V}r_{V, U}\qquad {\rm for} \qquad W\subset V\subset U\,.$$

. Such a structure is called a presheaf with values in $\mathcal{c}$.