Holmstrom Monoidal model category

A monoidal MC is like a ring in the 2-category of MCs.

Examples: Simplicial sets, pointed simplicial sets, chain complexes of modules over a commutative ring, chain complexes of comodules over a commutative Hopf algebra, k-spaces, compactly generated topological spaces. Non-example: Topological spaces.

Def: A monoidal category is a closed category $C$, with a model structure, such that the following holds:

1. The monoidal structure is a Quillen bifunctor
2. Let $q: QS \to S$ be a cofibrant replacement for the unit $S$. Then the natural map $q \otimes 1: QS \otimes x \to S \otimes X$ is a WE for all cofibrant $X$, and same for $1 \otimes q$. (This is automatic if $S$ is cofibrant.)

Similarly, can define symmetric monoidal model category.

Example: $Sset$ is a symmetric monoidal model category, and same for $Sset_*$. For any (symmetric) monoidal model category with cofibrant unit coinciding with the terminal object, the model category $C_*$ is also (symmetric) monoidal. CG spaces and k-spaces form symmetric monoidal model cats under the k-space product, and same for the pointed versions. Unbounded chain complexes of R-mods is a symmetric monoidal model category (using the standard model structure, not the injective one). Chain complexes of comodules over a commutative Hopf algebra over a field also form a SMMC. The model category R-mod is monoidal, if R is a quasi-Frobenius ring and a finitedimensional Hopf algebra over a field. It is symmetric iff R is cocommutative.

In the above examples the unit is cofibrant. The category of S-modules is a monoidal model category in which the unit is not cofibrant.

Hovey: “It is essential that $C$ is a closed monoidal category for $C_*$ to be monoidal. For example, the smash product on pointed topological spaces fails to be associative”. “$Top$ is not a monoidal model category, because it is not a closed monoidal category”.

Given two monoidal MCs, a monoidal Quillen adjunction between them is a Quillen adjunction $(F, U, \phi)$ such that $F$ is a monoidal functor, and $F$ applied to cofibrant replacement of the unit is a WE. The last condition is redundant when the unit is cofibrant, but necessary in general to ensure that the unit isomorphism passes to the homotopy category. The left adjoint above is referred to as a monoidal Quillen functor. We get a 2-category of monoidal MCs, and can also construct the 2-category of symmetric monoidal MCs, with symmetric monoidal Quillen functors.

Example: Adding a disjoint basepoint is a symmetric monoidal Quillen functor from $Sset$ to $Sset_*$, whose right adjoint is the forgetful functor. Similarly for k-spaces and CG spaces. Geometric realization is a symmetric monoidal Quillen functor from (pointed) simplicial sets to (pointed) k-spaces.

The homotopy category

Recall (Hovey p. 106) the definition of an adjunction in two variables.

Theorem: Suppose $C$ is a (symmetric) monoidal model category. Then its homotopy category can be given the structure of a closed (symmetric) monoidal category. The adjunction of two variables $(\otimes^L, RHom_r, RHom_l)$ that is part of the closed structure on $Ho \ C$ is the total derived adjunction of $(\otimes, Hom_r, Hom_l)$. The associativity, unit (and commutativity) IMs on $Ho \ C$ are derived from the corresponding IMs of $C$

We make this more precise, and give the statement for the more structured monoidal model cats. Let $C$ be a fixed monoidal model category.

Thm: The pseudo-2-functor $Ho$ from model cats to $Cat_{ad}$ lifts to a pseudo-2-functor:

• from monoidal model cats to closed monoidal cats
• from symmetric monoidal model cats to closed symmetric monoidal cats
• from $C$-model cats to closed $Ho \ C$-modules
• from monoidal $C$-model cats to closed $Ho \ C$-algebras
• from symmetric (resp. central) monoidal $C$-model cats to closed symmetric (resp. central) $Ho \ C$-algebras (if $C$ is symmetric)

nLab page on Monoidal model category