Unique Path Lifting from Homotopy Point of View

Abstract

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.

On h-Fibrations

“…It is straightforward that composition of two maps with wuphl is a map with wuphl [7,Proposition 4.1] and so we have the following proposition. Now, we can define category of h-fibrations, hFib and its subcategory, category of h-fibrations with wuphl, hFibwu in which the objects are topological spaces and morphisms are h-fibrations and h-fibrations with wuphl, respectively.…”

“…In Section 3, by proving that the composition of h-fibrations is an h-fibration, we introduce some new categories by h-fibrations hFib, hFibu and hFibwu. Then we compare them by the categories constructed by fibrations Fib, Fibu and Fibwu (see [7,8]). Moreover, we show that these new categories have products and coproducts by introducing them.…”

“…Unique path lifting property is important in the study of fibrations because make them a covering map, when the base space is locally nice, i.e, locally path connected and semi-locally simply connected [8]. The authors introduced a homotopical version of upl in [7] and studied its role in fibrations. Since here we are working with a weak homotopical version of fibrations, we are going to study h-fibrations with the homotopical version of upl.…”

“…Also, note that in the above diagram, the inclusions are proper. The first row is proper [7,Example 3.3] and the second row is proper, since a fibration is an h-fibration. Moreover, Example 2.12 shows that the second and the third column are proper.…”

“…Unique path lifting property has an important role for fibrations, because it makes them very close to covering projections and also implies lifting theorem [8,Theorem 2.4.5]. In [7], the authors presented a homotopical version of unique path lifting property and studied its properties for fibrations.…”