Enriched model categories and an application to additive endomorphism spectra

TL;DR

This paper introduces additive model categories, establishes their enrichment over symmetric spectra, and develops enriched model category theory, including adjoint modules and enrichment transfer results.

Contribution

It defines additive model categories, proves their enrichment over symmetric spectra, and develops the theory of adjoint modules and enrichment transfer.

Findings

Additive model categories can be enriched over symmetric spectra.

Objects in such categories have associated endomorphism spectra.

The paper develops the theory of adjoint modules and enrichment transfer.

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.