It is known that there is a wide class of path-connected topological spaces , which are not semilocally simply-connected but have a generalized universal covering, that is, a surjective map which is characterized by the usual unique lifting criterion and the fact that is path-connected, locally path-connected and simply-connected.
For a path-connected topological space and a map , we form the pullback of such a generalized universal covering and consider the following question: given a path-component of , when exactly is a generalized universal covering? We show that the classical criterion, of being injective, is too coarse a notion to be sufficient in this context and present its appropriate (necessary and sufficient) refinement.
"Pullbacks of generalized universal coverings." Algebr. Geom. Topol. 7 (3) 1379 - 1388, 2007. https://doi.org/10.2140/agt.2007.7.1379