The category of long exact sequences and the homotopy exact sequence of modules

Abstract

The relative homotopy theory of modules, including the (module) homotopy exact sequence, was developed by Peter Hilton (1965). Our thrust is to produce an alternative proof of the existence of the injective homotopy exact sequence with no reference to elements of sets, so that one can define the necessary homotopy concepts in arbitrary abelian categories with enough injectives and projectives, and obtain, automatically, the projective relative homotopy theory as the dual. Furthermore, we pursue the relative (module) homotopy theory analogously to the absolute (module) homotopy theory. For these purposes, we embed the relative category into the category of long exact sequences, as a full subcategory, in our search for suitable notions of monomorphisms and injectives in the relative category.