One equivalently (with non-negligible but conventional chance of confusion of terminology) calls such a -groupoid principal bundle.
So more generally, for any groupoid object with collection of objects, the -pullbacks
are groupoid principal bundles .
Lie groupoid principal bundles
For = and a Lie groupoid, a -prinipal bundle is locally of the form
for the target fiber over an object , or equivalently of the form .
More precisely, we have the following sub-definition for Lie groupoid principal bundles: Given a Lie groupoid , a -principal bundle over a manifold (with right action) is a surjective submersion together with a moment map (sometimes also called an anchor), such that there is a (right) -action on , that is, there is map , such that
Moreover, we require the -action to be free and proper, that is,
[free] is injective,
[proper] is a proper map , i.e. the preimage of a compact is compact.
These two conditions are equivalent to the fact that