nLab double category of model categories

Definition

Definition

(double category of model categories)

The (very large) double category of model categories $ModCat_{dbl}$ has

1. as objects: model categories $\mathcal{C}$;

2. as vertical morphisms: left Quillen functors $\mathcal{C} \overset{L}{\longrightarrow} \mathcal{E}$;

3. as horizontal morphisms: right Quillen functors $\mathcal{C} \overset{R}{\longrightarrow}\mathcal{D}$;

4. as 2-morphisms natural transformations between the composites of underlying functors.

   $$L_2\circ R_1 \overset{\phi}{\Rightarrow} R_2\circ L_1 \phantom{AAAAA} \array{ \mathcal{C} &\overset{\phantom{AA}R_1\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{C} &\underset{\phantom{AA}R_2\phantom{AA}}{\longrightarrow}& \mathcal{D} }$$

and composition is given by ordinary composition of functors, horizontally and vertically, and by whiskering-composition of natural transformations.

(Shulman 07, Example 4.6)

There is hence a forgetful double functor

$$F \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)$$

to the double category of squares in the 2-category of categories, which forgets the model category-structure and the Quillen functor-property.

There is also another double pseudofunctor to $Sq(Cat)$ of interest, this is Prop. below.

Properties

Proposition

(homotopy double pseudofunctor on the double category of model categories)

There is a double pseudofunctor

$$Ho(-) \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)$$

from the double category of model categories (Def. ) to the double category of squares in the 2-category Cat, which sends

1. a model category $\mathcal{C}$ to its homotopy category of a model category;

2. a left Quillen functor to its left derived functor;

3. a right Quillen functor to its right derived functor;

4. a natural transformation

   $$\array{ \mathcal{C} &\overset{R_1}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{E} &\underset{R_2}{\longrightarrow}& \mathcal{F} }$$

   to the “derived natural transformation”

   $$\array{ Ho(\mathcal{C}) &\overset{\mathbb{R}R_1}{\longrightarrow}& Ho(\mathcal{D}) \\ {}^{\mathllap{\mathbb{L}L_1}}\Big\downarrow &\overset{Ho(\phi)}{\swArrow}& \Big\downarrow{}^{\mathrlap{\mathbb{L}L_2}} \\ Ho(\mathcal{E}) &\underset{\mathbb{R}R_2}{\longrightarrow}& Ho(\mathcal{F}) }$$

   given by the zig-zag

   (1)$$Ho(\phi) \;\colon\; L_2 Q R_1 P \overset{}{\longleftarrow} L_2 Q R_1 Q P \longrightarrow L_2 R_1 Q P \overset{\phi}{\longrightarrow} R_2 L_1 Q P \longrightarrow R_2 P L1 Q P \longleftarrow R_2 R L_1 Q \,,$$

   where the unlabeled morphisms are induced by fibrant resolution $c \to P c$ and cofibrant resolution $Q c \to c$, respectively.

(Shulman 07, Theorem 7.6)

Proposition

(recognizing derived natural isomorphisms)

For the derived natural transformation $Ho(\phi)$ in (1) to be invertible in the homotopy category, it is sufficient that for every object $c \in \mathcal{C}$ which is both fibrant and cofibrant the following natural transformation

$$R_2 Q L_1 c \overset{ R_2 p_{L_1 c} }{\longrightarrow} R_2 L_1 c \overset{\phi}{\longrightarrow} L_2 R_1 c \overset{ L_2 j_{R_1 c} }{\longrightarrow} L_2 P R_1 c$$

is invertible in the homotopy category, hence that the composite is a weak equivalences (by this Prop.).

(Shulman 07, Remark 7.2)

Examples

Example

(derived functor of left-right Quillen functor)

Let $\mathcal{C}$, $\mathcal{D}$ be model categories, and let

$$\mathcal{C} \overset{\phantom{A}F\phantom{A}}{\longrightarrow} \mathcal{C}$$

be a functor that is both a left Quillen functor as well as a right Quillen functor. This means equivalently that there is a 2-morphism in the double category of model categories (Def. ) of the form

(2)$$\array{ \mathcal{C} &\overset{\phantom{AA}F\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{F}}\Big\downarrow &{}^{id}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{D} &\underset{\phantom{A}id\phantom{A}}{\longrightarrow}& \mathcal{D} }$$

It follows that the left derived functor $\mathbb{L}F$ and right derived functor $\mathbb{R}F$ of $F$ are naturally isomorphic:

$$Ho(\mathcal{C}) \overset{ \mathbb{L}F \simeq \mathbb{R}F }{\longrightarrow} Ho(\mathcal{D}) \,.$$

(Shulman 07, corollary 7.8)

Proof

To see the natural isomorphism $\mathbb{L}F \simeq \mathbb{R}F$: By Prop. this is implied once the derived natural transformation $Ho(id)$ of (2) is a natural isomorphism. By Prop. this is the case, in the present situation, if the composition of

$$Q F c \overset{ p_{F c} }{\longrightarrow} F c \overset{ j_{F c} }{\longrightarrow} P F c$$

is a weak equivalence. But this is immediate, since the two factors are weak equivalences, by definition of fibrant/cofibrant resolution.

Related concepts

• 2-category of model categories

• 2Ho(CombModCat)

References

• Michael Shulman, Comparing composites of left and right derived functors, New York Journal of Mathematics 17 (2011), 75-125 (arXiv:0706.2868)