[Paper Review] Monoidal model categories

Mark Hovey|ArXiv.org|Feb 28, 1998

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper establishes a homotopy category of monoids and modules in cofibrantly generated monoidal model categories with cofibrant unit, even when Schwede-Shipley's conditions fail. It proves that the homotopy category is well-defined and homotopy invariant, extending homotopy theory to monoidal structures in topological settings like symmetric spectra.

ABSTRACT

A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a monoidal model category, one can consider monoids and modules over a given monoid. We would like to be able to study the homotopy theory of these monoids and modules. This question was first addressed by Stefan Schwede and Brooke Shipley in "Algebras and modules in monoidal model categories", who showed that under certain conditions, there are model categories of monoids and of modules over a given monoid. This paper is a follow-up to that one. We study what happens when the conditions of Schwede-Shipley do not hold. This will happen in any topological situation, and in particular, in topological symmetric spectra. We find that, with no conditions on our monoidal model category except that it be cofibrantly generated and that the unit be cofibrant, we still obtain a homotopy category of monoids, and that this homotopy category is homotopy invariant in an appropriate sense.

Motivation & Objective

• To extend homotopy theory of monoids and modules beyond the constraints of Schwede-Shipley's original conditions.
• To address the challenge of defining a homotopy category of monoids in monoidal model categories where the monoidal structure does not satisfy standard fibrancy or cofibrancy assumptions.
• To ensure the resulting homotopy category is homotopy invariant and well-behaved in topological contexts such as symmetric spectra.
• To provide a general framework applicable to topological symmetric spectra and other monoidal model categories where the unit is cofibrant but other conditions fail.

Proposed method

• Adopting a cofibrantly generated monoidal model category with cofibrant unit as the foundational setting.
• Constructing a homotopy category of monoids via a localization process that respects the monoidal structure.
• Using the cofibrant unit condition to ensure compatibility between the monoidal structure and weak equivalences.
• Defining modules over a monoid in the homotopy category and proving their homotopy invariance.
• Establishing that the resulting homotopy category is independent of choices and respects the monoidal operations up to homotopy.
• Applying the framework to topological symmetric spectra, where standard conditions fail but the construction still yields a well-defined homotopy theory.

Experimental results

Research questions

• RQ1 Can a homotopy category of monoids be defined in a monoidal model category when Schwede-Shipley's conditions do not hold?
• RQ2 Is the homotopy category of monoids homotopy invariant under weak equivalences in the underlying category?
• RQ3 How does the cofibrant unit condition enable the construction of a well-behaved homotopy category of monoids?
• RQ4 Can this framework be applied to topological symmetric spectra, where standard conditions fail?
• RQ5 What structural properties ensure the homotopy category of modules over a monoid is well-defined and invariant?

Key findings

• A homotopy category of monoids exists in any cofibrantly generated monoidal model category with cofibrant unit, even without Schwede-Shipley's additional conditions.
• The homotopy category of monoids is homotopy invariant, meaning it respects weak equivalences and is independent of choices in the construction.
• The construction extends to modules over a monoid, yielding a well-defined homotopy category of modules.
• The framework applies to topological symmetric spectra, where the monoidal structure does not satisfy the standard conditions but the homotopy theory remains well-behaved.
• The cofibrant unit condition is sufficient to ensure the existence and invariance of the homotopy category, even in the absence of other technical assumptions.