Arrow Categories of Monoidal Model Categories

TL;DR

This paper proves that the arrow category of a monoidal model category, with the pushout product structure, is itself a monoidal model category, enabling new applications in cubical homotopy theory.

Contribution

It establishes that the arrow category inherits a monoidal model structure, answering an open question and broadening the scope of monoidal model categories.

Findings

Arrow categories of monoidal model categories are themselves monoidal model categories.

Includes examples like chain complexes, small categories, topological spaces, and pro-categories.

Enables monoidal product considerations in cubical homotopy theory.

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.