Showing changes from revision #1 to #2:
Added | Removed | Changed
Let be a monoidal model category.
A functor in is a weak equivaleence if the induced functor is an equivalence of -enriched categories.
In other words: F is a weak equivalence iff
(1) For every pair , the induced map
is a weak equivalence in .
(2) is essentially surjective on the level of homotopy categories.
Recall that the homotopy category of a model category was defined to have the same objects as and the objects consist of the equivalence classes of the morphisms in the objects wrt. the homotopy equivalence relation. Here two morphisms of were called to be homotopic if their product map factors through some path object of their codomain.
The following definition says a functor between categories is called a quasi fibrations if every isomorphism has a lift with respect to .
Let be a monoidal model category. Let denote the category of small -enriched categories.
Given a monoidal structure on also its homotopy category (this was defined to have the same objects and the objects consist of the equivalence classes of the morphisms in the objects wrt. homotopy
are those of carries a monoidal structure which is determined up to a unique isomorphism by the requirement that there exists a monoidal functor
from to its homotopy category.
Let :C\to D$ be a functor monoidal between model classical category. categories.
A functor is called a quasi-fibration in if, for every object is a weak equivaleence if the induced functor and every isomorphism is an equivalence of in -enriched categories., there exists an isomorphism in such that .
In other words: F is a weak equivalence iff
(1) For every pair , the induced map
is a weak equivalence in .
(2) is essentially surjective on the level of homotopy categories.
Let be an excellent model category. Then:
An -enriched category is a fibrant object of iff it is locally fibrant: i.e. for all the hom object is fibrant.
Let be a -enriched functor where is a fibrant object of . Then is a fibration iff is a local fibration.
The following definition says a functor between categories is called a quasi fibrations if every isomorphism has a lift with respect to .
Let :C\to D$ be a monoidal functor category. between Let classical categories. be an -enriched category.
(1) A morphism is called a in quasi-fibration if, for every object is called an equivalence and every isomorphism if the homotopy class in of , there exists an isomorphism is an isomorphism in in such that .
(2) is called locally fibrant object if for every pair of objects , the mapping space is a fibrant object of .
(3) An -enriched functor is called a local fibration if the following conditions are satisfied:
(3.i) is a fibration in for every .
(3.ii) The induced map is a quasi-fibration of categories.
(excellent Let model category) be an excellent model category. Then:
A model category is called excellent model category if it is equipped with a symmetric monoidal structure and satisfies the following conditions
An -enriched category is a fibrant object of iff it is locally fibrant: i.e. for all the hom object is fibrant.
Let be a -enriched functor where is a fibrant object of . Then is a fibration iff is a local fibration.
(A1) is combinatorial.
(A2) Every monomorphism in is a cofibration and the collection of cofibrations in is stable under products.
(A3) The collection of weak equivalencies in is stable under filtered colimits.
(A4) is a Quillen bifunctor.
(A5) The monoidal model category satisfies the invertibility hypothesis.
Let be a monoidal category. Let be an -enriched category.
(1) A morphism in is called an equivalence if the homotopy class of is an isomorphism in .
(2) is called locally fibrant object if for every pair of objects , the mapping space is a fibrant object of .
(3) An -enriched functor is called a local fibration if the following conditions are satisfied:
(3.i) is a fibration in for every .
(3.ii) The induced map is a quasi-fibration of categories.
(excellent model category)
A model category is called excellent model category if it is equipped with a symmetric monoidal structure and satisfies the following conditions
(A1) is combinatorial.
(A2) Every monomorphism in is a cofibration and the collection of cofibrations in is stable under products.
(A3) The collection of weak equivalencies in is stable under filtered colimits.
(A4) is a Quillen bifunctor.
(A5) The monoidal model category satisfies the invertibility hypothesis.