A stack is presented by a groupoid , if it is the stack (then usually regarded as a stack on the siteTop or Diff) which assigns to each test domain the category of -principal bundles over
Formulation in terms of nonabelian cohomology
This can be reformulated as follows: for a manifold let denote the internal hom of groupoids (or of categories with topological or smooth structure), with regarded as the discrete groupoid over . We can regard this as the groupoid of trivial -principal bundles over :
This is contravariantly functorial in and indeed yields a groupoid-valued presheaf
the presheaf of trivial -principal bundles.
So the stack presented by is the stackification of this groupoid-valued presheaf.
In yet other words this means nothing but that the stack presented by is the nonabelian cohomology with coefficients in :