Editing Gerbe (section)

== Definitions ==

===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.

=== Gerbes on a site ===
The most general definition of gerbes are defined over a [[Site (mathematics)|site]]. Given a site <math>\mathcal{C}</math> a <math>\mathcal{C}</math>-gerbe <math>G</math><ref name="stacks.math.columbia.edu">{{Cite web|title=Section 8.11 (06NY): Gerbes—The Stacks project|url=https://stacks.math.columbia.edu/tag/06NY|access-date=2020-10-27|website=stacks.math.columbia.edu}}</ref><ref>{{Cite book|last=Giraud, J. (Jean)|url=https://www.worldcat.org/oclc/186709|title=Cohomologie non abélienne.|date=1971|publisher=Springer-Verlag|isbn=3-540-05307-7|location=Berlin|oclc=186709}}</ref>{{rp|129}} is a category fibered in groupoids <math>G \to \mathcal{C}</math> such that

# There exists a refinement<ref>{{Cite web|title=Section 7.8 (00VS): Families of morphisms with fixed target—The Stacks project|url=https://stacks.math.columbia.edu/tag/00VS|access-date=2020-10-27|website=stacks.math.columbia.edu}}</ref> <math>\mathcal{C}'</math> of <math>\mathcal{C}</math> such that for every object <math>S \in \text{Ob}(\mathcal{C}')</math> the associated fibered category <math>G_S</math> is non-empty
# For every <math>S \in \text{Ob}(\mathcal{C})</math> any two objects in the fibered category <math>G_S</math> are locally isomorphic
Note that for a site <math>\mathcal{C}</math> with a final object <math>e</math>, a category fibered in groupoids <math>G \to \mathcal{C}</math> is a <math>\mathcal{C}</math>-gerbe admits a local section, meaning satisfies the first axiom, if <math>\text{Ob}(G_e) \neq \varnothing</math>.

==== 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.