Cofibration and Fibration Structures in Enriched Categories

Abstract:  This paper is motivated by the attempt to lay out the grounds for relative homological algebra for module spectra. Elmendorf, Kriz, Mandell, and May constructed an associative, commutative and unital smash product for spectra with some additional structure, called S-module spectra [6]. This good smash product allows the translation of concepts of classical homological algebra to the category S Mod of S-module spectra. Since S Mod is not abelian, the standard approach to derived functors does not apply. The most common frame work for doing homological algebra in the non-abelian case is Quillen's concept of a closed model category sytructure. Elmendorf, Kriz, Mandell and May construct a topological closed model structure on S Mod whose weak equivalences are maps inducing isomorphisms on homotopy groups and whose cofibrations are retracts of relative cell-objects. Relative cell-objects are constructed analogous to relative CW -complexes. Relative homological algebra deals with weak equivalences which are genuine homotopy equivalences after forgetting part of the structure. Genuine homotopy equivalences often cannot be tested by a "set of small objects", and in the absence of small objects arguments one has trouble to define closed model structures. Fortunately, one only needs parts of a closed model structure to define left or right derived functors: cofibration and fibration structures suffice [1], [13], and genuine homotopy equivalences fit nicely into such structures. One of the most useful technical results for cofibrations of topological spaces is the "product theorem": if i : A ! X and j : B ! Y are cofibrations and one of them is closed then (i \Theta Y; X \Theta j) : A \Theta Y [ X \Theta B