# 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.

http://www.tac.mta.ca/tac/volumes/18/15/18-15.dvi
http://www.tac.mta.ca/tac/volumes/18/15/18-15.ps
http://www.tac.mta.ca/tac/volumes/18/15/18-15.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/15/18-15.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/15/18-15.ps

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