Editing Gerbe (section)

== Cohomological classification ==
One of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups <math>\underline{L}</math>,<ref>{{Cite web|title=Section 21.11 (0CJZ): Second cohomology and gerbes—The Stacks project|url=https://stacks.math.columbia.edu/tag/0CJZ|access-date=2020-10-27|website=stacks.math.columbia.edu}}</ref><ref name="stacks.math.columbia.edu"/> called a band. For a gerbe <math>\mathcal{X}</math> on a site <math>\mathcal{C}</math>, an object <math>U \in \text{Ob}(\mathcal{C})</math>, and an object <math>x \in \text{Ob}(\mathcal{X}(U))</math>, the automorphism group of a gerbe is defined as the automorphism group <math>L = \underline{\text{Aut}}_{\mathcal{X}(U)}(x)</math>. Notice this is well defined whenever the automorphism group is always the same. Given a covering <math>\mathcal{U} = \{U_i \to X \}_{i \in I}</math>, there is an associated class<blockquote><math>c(\underline{L}) \in H^3(X,\underline{L})</math></blockquote>representing the [[isomorphism class]] of the gerbe <math>\mathcal{X}</math> banded by <math>L</math>.

For example, in topology, many examples of gerbes can be constructed by considering gerbes banded by the group <math>U(1)</math>. As the classifying space <math>B(U(1)) = K(\mathbb{Z},2)</math> is the second [[Eilenberg–MacLane space|Eilenberg–Maclane]] space for the integers, a bundle gerbe banded by <math>U(1)</math> on a topological space <math>X</math> is constructed from a homotopy class of maps in<blockquote><math>[X, B^2(U(1))] = [X,K(\mathbb{Z},3)]</math>,</blockquote>which is exactly the third [[singular homology]] group <math>H^3(X,\mathbb{Z})</math>. It has been found<ref>{{cite arXiv|last=Karoubi|first=Max|date=2010-12-12|title=Twisted bundles and twisted K-theory|class=math.KT|eprint=1012.2512}}</ref> that all gerbes representing torsion cohomology classes in <math>H^3(X,\mathbb{Z})</math> are represented by a bundle of finite dimensional algebras <math>\text{End}(V)</math> for a fixed complex vector space <math>V</math>. In addition, the non-torsion classes are represented as infinite-dimensional principal bundles <math>PU(\mathcal{H})</math> of the projective group of unitary operators on a fixed infinite dimensional [[Separable space|separable]] [[Hilbert space]] <math>\mathcal{H}</math>. Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square-summable sequences <math>\ell^2</math>.

The homotopy-theoretic interpretation of gerbes comes from looking at the [[homotopy fiber square]]<blockquote><math>\begin{matrix}
\mathcal{X} & \to & * \\
\downarrow &  & \downarrow \\
S & \xrightarrow{f} & B^2U(1)
\end{matrix}</math></blockquote>analogous to how a line bundle comes from the homotopy fiber square<blockquote><math>\begin{matrix}
L & \to & * \\
\downarrow &  & \downarrow \\
S & \xrightarrow{f} & BU(1)
\end{matrix}</math></blockquote>where <math>BU(1) \simeq K(\mathbb{Z},2)</math>, giving <math>H^2(S,\mathbb{Z})</math> as the group of isomorphism classes of line bundles on <math>S</math>.