Unique Path Lifting from Homotopy Point of View

TL;DR

This paper introduces new homotopical notions extending the unique path lifting property, explores their relationships in fibrations, and examines their implications for the structure of fibrations and fundamental groups.

Contribution

It defines and analyzes homotopical variants of the unique path lifting property, establishing their relationships and impact on fibrations and fundamental groups.

Findings

Established relationships between new homotopical notions and fibrations.

Proved existence of products and coproducts in categories with these notions.

Compared new fibrations categories to generalized covering map categories.

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.