Unique Path Lifting from Homotopy Point of View

On h-Fibrations

“…The unique path lifting property for fibrations is equivalent to the fact that every path in any fiber is constant [8, Theorem 2.2.5]. Also, the weakly unique path homotopically lifting property for fibrations is equivalent to the fact that every loop in any fiber is nullhomotopic [7,Theorem 3.4]. In the following, we show that these facts hold for h-fibrations.…”

“…Unique path lifting property has an important role for fibrations, because make 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 it's properties for fibrations.…”

“…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.…”

“…A map p : E → B is said to have weakly unique path homotopically lifting property abbreviated by wuphl, if by given two paths α and β in E with α(0) = β(0), α(1) = β(1) and p • α ≃ p • β, rel İ, then it follows that α ≃ β, rel İ (see [7]). The unique path lifting property for fibrations is equivalent to the fact that every path in any fiber is constant [8, Theorem 2.2.5].…”