Reedy diagrams in V-model categories

TL;DR

This paper investigates the structure of Reedy diagrams within V-model categories, establishing conditions under which diagram categories inherit monoidal and model category structures, thus advancing the understanding of diagrammatic homotopy theory.

Contribution

It provides explicit conditions for diagram categories to be closed symmetric monoidal and V-model categories, extending Reedy diagram theory in enriched model categories.

Findings

V^K is a closed symmetric monoidal category when K is small and V is closed symmetric monoidal.

C^K becomes a closed V^K-module under the given conditions.

V^K forms a monoidal model category if K is Reedy, V is monoidal, and C is a V-model category.

Abstract

We study the category of Reedy diagrams in a $\mm$-model category. Explicitly, we show that if K is a small category, V is a closed symmetric monoidal category and C is a closed V-module, then the diagram category V^K is a closed symmetric monoidal category and the diagram category C^K is a closed V^K-module. We then prove that if further K is a Reedy category, V is a monoidal model category and C is a V-model category, then with the Reedy model category structures, V^K is a monoidal model category and C^K$ is a $\mm^K-model category provided that either the unit 1 of V is cofibrant or V is cofibrantly generated.