Editing Gerbe (section)

===Gerbes on a topological space===

A gerbe on a [[topological space]] <math>S</math><ref>{{Cite book|url=https://www.worldcat.org/oclc/233973513|title=Basic bundle theory and K-cohomology invariants|date=2008|publisher=Springer|others=Husemöller, Dale.|isbn=978-3-540-74956-1|location=Berlin|oclc=233973513}}</ref>{{rp|318}} is a [[stack (mathematics)|stack]] <math>\mathcal{X}</math> of [[groupoid]]s over <math>S</math> that is ''locally non-empty'' (each point <math>p \in S</math> has an open neighbourhood <math>U_p</math> over which the [[Section (category theory)|section category]] <math>\mathcal{X}(U_p)</math> of the gerbe is not empty) and ''transitive'' (for any two objects <math>a</math> and <math>b</math> of <math>\mathcal{X}(U)</math> for any open set <math>U</math>, there is an open covering <math>\mathcal{U} = \{U_i \}_{i \in I}</math> of <math>U</math> such that the restrictions of <math>a</math> and <math>b</math> to each <math>U_i</math> are connected by at least one morphism).

A canonical example is the gerbe <math>BH</math> of [[principal bundles]] with a fixed [[structure group]] <math>H</math>: the section category over an open set <math>U</math> is the category of principal <math>H</math>-bundles on <math>U</math> with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle <math>X \times H \to X</math> shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.