model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -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 has
as objects: model categories ;
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 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 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 and cofibrant resolution , respectively.
(recognizing derived natural isomorphisms)
For the derived natural transformation in (1) to be invertible in the homotopy category, it is sufficient that for every object 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 , 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 and right derived functor of are naturally isomorphic:
To see the natural isomorphism : By Prop. this is implied once the derived natural transformation 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.