Reedy diagrams in symmetric monoidal model categories

TL;DR

This paper demonstrates that diagram categories over Reedy categories inherit symmetric monoidal model structures under certain conditions, extending the framework for structured diagrammatic homotopy theory.

Contribution

It establishes that diagram categories over Reedy categories are symmetric monoidal model categories when the base category has a compatible model structure and is cofibrantly generated.

Findings

Diagram categories over small categories are symmetric monoidal.

Reedy model structures are compatible with monoidal structures under certain conditions.

The results extend the applicability of Reedy categories in homotopy theory.

Abstract

Given a small category $I$ and a closed symmetric monoidal category $\mm$, we show that the diagram category $\mm^I$ with the objectwise product is a closed symmetric monoidal category. We then prove that if $I$ is a Reedy category and $\mm$ has a model structure compatible with its product, then so is the Reedy model structure on $\mm^I$ provided that $\mm$ is cofibrantly generated.