  The paper is devoted to introduce some notions extending the unique path
lifting property from a homotopy viewpoint and to study their roles in the
category of fibrations. First, we define some homotopical kinds of the unique
path lifting property and find all possible relationships between them.
Moreover, we supplement the full relationships of these new notions in the
presence of fibrations. Second, we deduce some results in the category of
fibrations with these notions instead of unique path lifting such as the
existence of products and coproducts. Also, we give a brief comparison of these
new categories to some categories of the other generalizations of covering
maps. Finally, we present two subgroups of the fundamental group related to the
fibrations with these notions and compare them to the subgroups of the
fundamental group related to covering and generalized covering maps.
