Categories and General Algebraic Structures with Applications

Categories and General Algebraic Structures with Applications

$\mathcal{H}$-Fibrations‎: ‎Fibrations in Homotopy Category

Document Type : Research Paper

Authors
Ali Pakdaman 1
Saba Dehrooye 1
Mehdi Tajik 2
Behrooz Mashayekhy 2
1 Department of Mathematics‎, ‎Faculty of Science‎, ‎Golestan university, ‎P.O.Box 155‎, ‎Gorgan‎, ‎Iran
2 Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran.
10.48308/cgasa.2024.234981.1474
Abstract
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.
Keywords
Fiber homotopy
h-fibration
$\mathcal{H}$-fibration
homotopical path lifting
Subjects
[1] Dold, A., Partitions of unity in the theory of fibrations, Annal. Math. 78 (1963), 223-255.
[2] Fuchs, K., “Homotopiefaserungen”, Dissertation, Universit¨at des Saarbr¨ucken.
[3] Paveˇsi´c, P., A note on trivial fibrations, Glas. Mat. 46 (2011), 513-519.
[4] Paveˇsi´c, P. and Piccinini, R.A., “Fibrations and their Classification”, Heldermann Verlag, 2013.
[5] Tajik, M., Mashayekhy, B., and Pakdaman, A., Unique path lifting from homotopy point of view, Acta Math. Vietnam. 43(2) (2018), 257-273.
[6] Tajik, M., Mashayekhy, B., and Pakdaman, A., h-fibrations, Hacet. J. Math. Stat. 48(3) (2019), 732-742.
[7] Spanier, E.H., “Algebraic Topology”, McGraw-Hill, 1966.