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The category of necklaces is Reedy monoidal

Violeta Borges Marques, Arne Mertens

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.

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 which is monoidal such that for all symmetric monoidal model categories , the category is model monoidal when equipped with the Day convolution. In the second part, we study the category 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 is simple Reedy monoidal.
Paper Structure (8 sections, 20 theorems, 16 equations, 2 tables)

This paper contains 8 sections, 20 theorems, 16 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Reedy monoidal categories
  4. Reedy categories and right fibrations
  5. Reedy monoidal model structures
  6. The category of necklaces
  7. A combinatorial description of necklaces
  8. The Reedy monoidal structure on necklaces

Key Result

Proposition 2.9

Let $F: \mathcal{R}\rightarrow \mathcal{S}$ be a morphisms of Reedy categories. If the restriction $F^{\to}: \mathcal{R}^{\to}\rightarrow \mathcal{S}^{\to}$ is a discrete fibration of categories, then $F: \mathcal{R}\rightarrow \mathcal{S}$ is a right fibration of Reedy categories.

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 2.1: hovey1999model, Definition 5.2.1
  • Definition 2.3: hovey1999model, Def 5.1.2
  • Remark 2.4
  • Definition 2.5: Barwick, Definition 3.16.1
  • Definition 2.6: Barwick, Theorem 3.22
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 47 more