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Arrow Categories of Monoidal Model Categories

David White, Donald Yau

Abstract

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, and has the important consequence that it allows for the consideration of a monoidal product in cubical homotopy theory. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, topological spaces, and pro-categories.

Arrow Categories of Monoidal Model Categories

Abstract

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, and has the important consequence that it allows for the consideration of a monoidal product in cubical homotopy theory. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, topological spaces, and pro-categories.

Paper Structure

This paper contains 12 sections, 9 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Projective Model Structure on the Arrow Category
  3. Arrow Category as a Monoidal Model Category
  4. Examples : Non-Cofibrantly Generated Monoidal Model Categories
  5. Christensen-Hovey Model Structure on Integral Chain Complexes
  6. Barthel-May-Riehl Model Structure on DG-Modules
  7. Adámek-Herrlich-Rosický-Tholen Model Structure on Small Categories
  8. Strøm Model Structure on Compactly Generated Spaces
  9. Pro-categories
  10. On presentable $\infty$-categories
  11. Application to Cubical Homotopy Theory
  12. Cubical Diagram Categories of Monoidal Model Categories

Key Result

Theorem A

Suppose $\mathsf{M}$ is a monoidal model category. Then its arrow category equipped with the pushout product monoidal structure and the projective model structure is a monoidal model category.

Theorems & Definitions (19)

  • Theorem A
  • Corollary B
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Lemma 3.5
  • proof
  • ...and 9 more