model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
(double category of model categories)
The (very large) double category of model categories $ModCat_{dbl}$ has
as objects: model categories $\mathcal{C}$;
as vertical morphisms: left Quillen functors $\mathcal{C} \overset{L}{\longrightarrow} \mathcal{E}$;
as horizontal morphisms: right Quillen functors $\mathcal{C} \overset{R}{\longrightarrow}\mathcal{D}$;
as 2-morphisms natural transformations between the composites of underlying functors.
and composition is given by ordinary composition of functors, horizontally and vertically, and by whiskering-composition of natural transformations.
There is hence a forgetful double functor
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.
(homotopy double pseudofunctor on the double category of model categories)
There is a double pseudofunctor
from the double category of model categories (Def. ) to the double category of squares in the 2-category Cat, which sends
a model category $\mathcal{C}$ to its homotopy category of a model category;
a left Quillen functor to its left derived functor;
a right Quillen functor to its right derived functor;
to the “derived natural transformation”
given by the zig-zag
where the unlabeled morphisms are induced by fibrant resolution $c \to P c$ and cofibrant resolution $Q c \to c$, respectively.
(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
is invertible in the homotopy category, hence that the composite is a weak equivalences (by this Prop.).
(derived functor of left-right Quillen functor)
Let $\mathcal{C}$, $\mathcal{D}$ be model categories, and let
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
It follows that the left derived functor $\mathbb{L}F$ and right derived functor $\mathbb{R}F$ of $F$ are naturally isomorphic:
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
is a weak equivalence. But this is immediate, since the two factors are weak equivalences, by definition of fibrant/cofibrant resolution.
Last revised on May 6, 2023 at 15:46:17. See the history of this page for a list of all contributions to it.