Editing Gerbe (section)

==== Motivation for gerbes on a site ====
One of the main motivations for considering gerbes on a site is to consider the following naive question: if the Cech cohomology group <math>H^1(\mathcal{U},G)</math> for a suitable covering <math>\mathcal{U} = \{U_i\}_{i \in I}</math> of a space <math>X</math> gives the isomorphism classes of principal <math>G</math>-bundles over <math>X</math>, what does the iterated cohomology functor <math>H^1(-,H^1(-,G))</math> represent? Meaning, we are gluing together the groups <math>H^1(U_i,G)</math> via some one cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group <math>H^2(\mathcal{U},G)</math>. It is expected this intuition should hold for [[higher gerbe]]s.