# nLab abelian model category

## Idea

An abelian model category is an abelian category with a compatible model structure.

## Definition

An abelian model category is an abelian category $\mathcal{A}$ that is complete and cocomplete, together with a model structure such that

## Relation to cotorsion pairs

Hovey has shown that, roughly speaking, model structures on abelian categories correspond to cotorsion pairs.

In one direction we have

###### Proposition

Let $\mathcal{A}$ be an abelian model category, i.e. an abelian category with a compatible model structure. Let $\mathcal{C}$, $\mathcal{F}$, and $\mathcal{W}$ denote the classes of cofibrant, fibrant, and trivial objects, respectively.

Then $(\mathcal{C} \cap \mathcal{W}, \mathcal{F})$ and $(\mathcal{C}, \mathcal{F} \cap \mathcal{W})$ are complete cotorsion pairs.

And under some more assumptions we have a converse

###### Theorem

Let $\mathcal{A}$ be an abelian category that is complete and cocomplete. Let $\mathcal{C}$, $\mathcal{F}$, and $\mathcal{W}$ denote three classes of objects in $\mathcal{A}$, such that $\mathcal{W}$ is a thick subcategory, and $(\mathcal{C} \cap \mathcal{W}, \mathcal{F})$ and $(\mathcal{C}, \mathcal{F} \cap \mathcal{W})$ are complete cotorsion pairs.

Then there exists a unique abelian model structure on $\mathcal{A}$ such that $\mathcal{C}$, $\mathcal{F}$, $\mathcal{W}$ are the classes of cofibrant, fibrant, and trivial objects, respectively.

Under certain assumptions on the cotorsion pair we can further guarantee that the associated model structure is monoidal.

###### Theorem

Under the assumptions of the previous theorem, suppose further that $\mathcal{A}$ is closed symmetric monoidal, and that

• Every object in the class $\mathcal{C}$ is flat? ($X \otimes \cdot$ is an exact functor).
• For any two objects $X$ and $Y$ in $\mathcal{C}$, the tensor product $X \otimes Y$ is also in $\mathcal{C}$. If one of $X$ and $Y$ is further in $\mathcal{W}$ then $X \otimes Y$ is also in $\mathcal{W}$.
• The unit is in $\mathcal{C}$.

Then $\mathcal{A}$ is a monoidal model category (with the model structure given by the previous theorem).

• Mark Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553–592. MR 2003m:55027

An overview is in