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

• (AMC1) A morphism $i : A \to B$ is a cofibration if and only if it is a monomorphism with cofibrant cokernel.
• (AMC2) A morphism $p : X \to Y$ is a fibration if and only if it is an epimorphism with fibrant kernel.

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).

References

• 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

• Mark Hovey, Cotorsion pairs and model categories, 2006 (pdf)