The category of necklaces is Reedy monoidal

TL;DR

This paper explores the structure of Reedy monoidal categories, focusing on the category of necklaces, and demonstrates that necklaces form a simple Reedy monoidal category, advancing the understanding of their combinatorial and categorical properties.

Contribution

It defines Reedy monoidal categories in the context of symmetric monoidal model categories and proves that the category of necklaces is a simple Reedy monoidal category.

Findings

Necklaces form a simple Reedy monoidal category

Streamlined proofs of existing results using combinatorial descriptions

Enhanced understanding of Reedy and monoidal interactions in categorical structures

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.