Enriched model categories and an application to additive endomorphism spectra

Abstract

We define the notion of an additive model category, and we prove that any additive, stable, combinatorial model category has a natural enrichment over symmetric spectra based on simplicial abelian groups. As a consequence, every object in such a model category has a naturally associated endomorphism ring inside this spectra category. We establish the basic properties of this enrichment. We also develop some enriched model category theory. In particular, we have a notion of an adjoint pair of functors being a 'module' over another such pair. Such things are called "adjoint modules". We develop the general theory of these, and use them to prove a result about transporting enrichments over one symmetric monoidal model category to a Quillen equivalent one.

Sections completely re-organized from previous version. Mathematical content all the same

References in corpus (2)

• Algebras and modules in monoidal model categories
• Equivalences of monoidal model categories

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• A Riemann--Hilbert correspondence for infinity local systems
• Levels of algebraicity in stable homotopy theories
• Monoidal cofibrant resolutions of dg algebras
• Homological Localisation of Model Categories