double category of model categories



Model category theory

model category



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2-Category theory




(double category of model categories)

The (very large) double category of model categories ModCat dblModCat_{dbl} has

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

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

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

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

    L 2R 1ϕR 2L 1AAAAA𝒞 AAR 1AA 𝒟 L 1 ϕ L 2 𝒞 AAR 2AA 𝒟 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:ModCat dblSq(Cat) 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)Sq(Cat) of interest, this is Prop. below.



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

There is a double pseudofunctor

Ho():ModCat dblSq(Cat) 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 𝒞\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

    𝒞 R 1 𝒟 L 1 ϕ L 2 R 2 \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

    Ho(𝒞) R 1 Ho(𝒟) 𝕃L 1 Ho(ϕ) 𝕃L 2 Ho() R 2 Ho() \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(ϕ):L 2QR 1PL 2QR 1QPL 2R 1QPϕR 2L 1QPR 2PL1QPR 2RL 1Q, 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 cPcc \to P c and cofibrant resolution QccQ c \to c, respectively.

(Shulman 07, Theorem 7.6)


(recognizing derived natural isomorphisms)

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

R 2QL 1cR 2p L 1cR 2L 1cϕL 2R 1cL 2j R 1cL 2PR 1c 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)



(derived functor of left-right Quillen functor)

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

𝒞AFA𝒞 \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)𝒞 AAFAA 𝒟 F id id 𝒟 AidA 𝒟 \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 𝕃F\mathbb{L}F and right derived functor F\mathbb{R}F of FF are naturally isomorphic:

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

(Shulman 07, corollary 7.8)


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

QFcp FcFcj FcPFc 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.


Last revised on October 13, 2021 at 10:47:03. See the history of this page for a list of all contributions to it.