Reedy diagrams in V-model categories (Q2336053)

This paper combines the theories of enriched category theory and Reedy model structures to prove the following result: Let \(\mathscr{V}\) be a closed symmetric monoidal model category and \(K\) a Reedy category. Suppose that the unit of \(\mathscr{V}\) is cofibrant or that \(\mathscr{V}\) is cofibrantly generated. Then \(\mathscr{V}^K\) with the objectwise product and the Reedy model structure is a closed symmetric monoidal model category. Furthermore, if \(\mathscr{C}\) is a \(\mathscr{V}\)-model category, then \(\mathscr{C}^K\) is a \(\mathscr{V}^K\)-model category with the objectwise action and the Reedy model structure. This combines results of \textit{B. Day} [Lect. Notes Math. 137, 1--38 (1970; Zbl 0203.31402)] in the enriched category direction and of Reedy's PhD thesis in the model category direction. Moreover, it generalizes a result of \textit{C. Barwick} [Homology Homotopy Appl. 12, No. 2, 245--320 (2010; Zbl 1243.18025)] which had additional assumptions on the Reedy category \(K\).