Homotopies of nullhomotopies in a module category I

Abstract

A relativization of the injective homotopy theory for modules over a ring R is presented, being the analogue for module homotopy of the homotopy category of ‘homotopy pairs’. This category provides an externalization of the phenomena associated with secondary composition. The basic properties of homotopy commutative squares are studied, the main tool being that of a difference element of a pair of nullhomotopies. A characterization of the isomorphisms of the relative homotopy category, known in the topological case, is shown to hold also for module homotopy.