Directed weak and directed Dold fibrations

Received: 4.V.2017, Accepted: 14.III.2018

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 the aforementioned article, introduces and studies the directed weak homotopy property (dWCHP) and directed  covering semistationary homotopy property (dCSHP) defining directed weak fibrations and directed Dold fibrations respectively, both extending to the category dTop the well known Dold’s [6] (or weak [3]) fibrations, but which are not equivalent as in the undirected case. Then the notion of a directed fibre homotopy equivalence (dFHE) between directed Dold  fibrations is studied. Some examples and counterexamples are given.
Keywords: Directed (d-) space, d- fibration, vertical d- homotopy, directed weak covering homotopy property, directed weak fibration, directed covering semistationary homotopy, directed Dold fibration, d-domination, semistationary d-homotopy, d-semistationary lifting pair, directed fibre homotopy equivalence, d-shrinkable

Mathematics Subject Classification (2010): 55R05, 55P99, 55U35, 55R65, 54E99

Author:

I. Pop, Faculty of Mathematics, “Al. I. Cuza” University, 700505-Iaşi, România