Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 24 1 2026 01 01 $\mathcal{H}$-Fibrations‎: ‎Fibrations in Homotopy Category 29 41 104940 10.48308/cgasa.2024.234981.1474 EN Ali Pakdaman Department of Mathematics‎, ‎Faculty of Science‎, ‎Golestan university, ‎P.O.Box 155‎, ‎Gorgan‎, ‎Iran 0000-0002-6174-1043 Saba Dehrooye Department of Mathematics‎, ‎Faculty of Science‎, ‎Golestan university, ‎P.O.Box 155‎, ‎Gorgan‎, ‎Iran Mehdi Tajik Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran. Behrooz Mashayekhy Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran. Journal Article 2024 03 04 In this paper we generalize fibrations by $\mathcal{H}$-fibrations, the maps which homotopically lift homotopies. We replace the equalities in the definition of covering homotopy property with the homotopy relation so that we can first get an expression of the concept of covering homotopy property in the homotopy category. After introducing $\mathcal{H}$-fibrations, we will have a homotopy expression of some concepts related to fibration, such as path lifting, lifting function and unique path lifting property, to generalize some results in fibration. In particular, we show that an $\mathcal{H}$-fibration has homotopical path lifting property and also prove that a map is an $\mathcal{H}$-fibration if and only if it has a homotopical lifting function. https://cgasa.sbu.ac.ir/article_104940_2dfc3a15689edfc90305c507af86e7e9.pdf