 Explicitly, given a space formula_49 with a formula_29-action, form the stack formula_68 which (intuitively speaking) sends a space formula_8 to the groupoid of pullback diagramsformula_79where formula_80 is a formula_29-equivariant morphism of spaces and formula_82is a principal formula_29-bundle.The morphisms in this category are just morphisms of diagrams where the arrows on the right-hand side are equal and the arrows on the left-hand side are morphisms of principal formula_29-bundles.A special case of this when "Y" is a point gives the classifying stack "BG" of a smooth affine group scheme "G"