It is known that there is a wide class of path-connected topological
spaces X, which are not semilocally simply-connected but have a
generalized universal covering, that is, a surjective map p
: ~X→ X which is characterized by the usual unique
lifting criterion and the fact that ~X is path-connected,
locally path-connected and simply-connected.
For a path-connected topological space Y and a map f: Y→
X, we form the pullback f*p: f* ~X→Y
of such a generalized universal covering p: ~X→X
and consider the following question: given a path-component ~Y
of f* ~X, when exactly is f* p∣~:
~Y→Y a generalized universal covering? We show that
the classical criterion, of f#: π1(Y)→π1(X)
being injective, is too coarse a notion to be sufficient in this context
and present its appropriate (necessary and sufficient) refinement.