Directed weak fibrations

TL;DR

This paper extends the theory of directed fibrations by introducing the directed weak homotopy property and directed weak fibrations within the category of directed spaces, providing new characterizations and examples.

Contribution

It introduces the concepts of directed weak homotopy property and directed weak fibrations, expanding the framework of homotopy theory in directed spaces.

Findings

Characterization of the directed weak homotopy property

Introduction of directed weak fibrations

Examples and counterexamples of directed weak fibrations

Abstract

In a previous paper [21] the author studied the homotopy lifting property in the category dTop of directed spaces in the sense of M. Grandis [12], [13], [14]. The present paper, which is a continuation of aforementioned article, introduces and studies the directed weak homotopy property (dWCHP) and directed weak fibrations, extending to the category dTop the well known Dold's (or weak) - fibrations [6]. The dWCHP is characterized in several ways. Then the notion of a directed fibre homotopy equivalence (dFHE) between directed weak fibrations is studied. Some examples and counterexamples are given.