The relative homotopy theory of modules, including the (module)homotopy exact sequence, was developed by Peter Hilton (1965). Ourthrust is to produce an alternative proof of the existence of theinjective homotopy exact sequence with no reference to elementsof sets, so that one can define the necessary homotopy concepts inarbitrary abelian categories with enough injectives andprojectives, and obtain, automatically, the projective relativehomotopy 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 relativecategory into the category of long exact sequences, as a fullsubcategory, in our search for suitable notions of monomorphismsand injectives in the relative category.
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