Enriched Model Categories and an Application to Additive Endomorphism Spectra

We define the notion of an additive model category and prove that any stable, additive, combinatorial model category M has a model enrichment over Sp(sAb) (symmetric spectra based on simplicial abelian groups). So to any object X ∈ M one can attach an endomorphism ring object, denoted hEndad(X), in this category of spectra. One can also obtain an associated differential graded algebra carrying the same information. We prove that the homotopy type of hEndad(X) is an invariant of Quillen equivalences between additive model categories. We also develop a general notion of an adjoint pair of functors being a ‘module’ over another such pair; we call such things adjoint modules. This is used to show that one can transport enrichments over one symmetric monoidal model category to a Quillen equivalent one, and in particular it is used to compare enrichments over Sp(sAb) and chain complexes.