Enriched model categories and an application to additive endomorphism spectra

Daniel Dugger and Brooke Shipley

We define the notion of an additive model category and prove that any stable, additive, combinatorial model category $\cal M$ has a model enrichment over $Sp^\Sigma(sAb)$ (symmetric spectra based on simplicial abelian groups). So to any object $X$ in $\cal M$ one can attach an endomorphism ring object, denoted $hEnd_ad(X)$, in the category $Sp^\Sigma(sAb)$. 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 $\Sp^\Sigma(s\Ab)$ and chain complexes.

Keywords: model categories, symmetric spectra, endomorphism ring

2000 MSC: 18D20, 55U35, 55P42, 18E05

Theory and Applications of Categories, Vol. 18, 2007, No. 15, pp 400-439.


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