Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The main result of Section 1 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. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section 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 being introduced here 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.