The category of necklaces is Reedy monoidal

In the first part of this note we further the study of the interactions between Reedy and monoidal structures on a small category, building upon the work of Barwick. We define a Reedy monoidal category as a Reedy category $\mathcal{R}$ which is monoidal such that for all symmetric monoidal model categories $\mathbf{\text{A}}$, the category $\mathrm{F}\mathrm{u}\mathrm{n}{({\mathcal{R}}^{\mathrm{o}\mathrm{p}},\mathbf{\text{A}})}_{\mathrm{R}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{y}}$ is model monoidal when equipped with the Day convolution. In the second part, we study the category $\mathcal{N}ec$ of necklaces, as defined by Baues and Dugger-Spivak. Making use of the combinatorial description present in arXiv:2302.02484v1, we streamline some proofs from the literature, and finally show that $\mathcal{N}ec$ is simple Reedy monoidal.