Stable homotopy monomorphisms
The concept of monomorphism in a category can be defined using an appropriate pullback diagram. But in the homotopy category of \({\mathcal T} {\mathcal O} {\mathcal P}\) pullbacks do not generally exist. Then \textit{M. Mather} [Can. J. Math. 28, 225-263 (1976; Zbl 0351.55005)] introduced another notion of homotopy pullbacks which do exist. Now the author of this paper modifies the notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The author shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology ``stable'' homotopy monomorphism. Also the author links these stable homotopy monomorphism to monomorphisms and products in the track homotopy category over a fixed space. In the second part of the paper the author studies whether a (weak) fibration is also a stable homotopy monomorphism. It is shown that the class of (weak) fibrations with this additional property coincides with the class of ``double'' (weak) fibrations. (The double (weak) covering homotopy property introduced in this paper is a stronger version of the \((W)CHP\) in which the final maps of the homotopies involved play the same role as the initial maps).
scientific article; zbMATH DE number 1036411