Enriched model categories and an application to additive endomorphism spectra

Abstract

We define the notion of an additive model category and prove that any stable, additive, combinatorial model category M has a model enrichment over SpΣ (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 the category SpΣ (sAb). We establish some useful properties of these endomorphism rings. We also develop a new notion in enriched category theory which we call 'adjoint modules'. This is used to compare enrichments over one symmetric monoidal model category with enrichments over a Quillen equivalent one. In particular, it is used here to compare enrichments over SpΣ (sAb) and chain complexes. © Daniel Dugger and Brooke Shipley, 2007.