The category of necklaces is Reedy monoidal

Abstract

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 $\textbf{A}$, the category $\mathrm{Fun}\left(\mathcal{R}^{\mathrm{op}}, \textbf{A}\right)_{\mathrm{Reedy}}$ 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.