nLab presented stack


A stack XX is presented by a groupoid GG, if it is the stack (then usually regarded as a stack on the site Top or Diff) which assigns to each test domain UU the category of GG-principal bundles over UU

X:=GBund():U{GprincipalbundlesonU}. X := G-Bund(-) : U \mapsto \{G-principal bundles on U\} \,.

Formulation in terms of nonabelian cohomology

This can be reformulated as follows: for XX a manifold let hom(X,G)hom(X,G) denote the internal hom of groupoids (or of categories with topological or smooth structure), with XX regarded as the discrete groupoid over XX. We can regard this as the groupoid of trivial GG-principal bundles over XX:

This is contravariantly functorial in XX and indeed yields a groupoid-valued presheaf

GTrivBund:Xhom(X,G) G-TrivBund : X \mapsto hom(X,G)

the presheaf of trivial GG-principal bundles.

So the stack presented by GG is the stackification of this groupoid-valued presheaf.

In yet other words this means nothing but that the stack presented by CC is the nonabelian cohomology H(,G)H(-,G) with coefficients in GG:

GBund():=H(,G). G-Bund(-) := H(-,G) \,.

Further resources

There is discussion of this and related aspects in Differential Nonabelian Cohomology in the private part of the nnLab.

Last revised on May 23, 2009 at 15:29:56. See the history of this page for a list of all contributions to it.