Reedy Diagrams in V-Model Categories

Abstract

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