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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.
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 functor between classical categories.
is called a quasi-fibration if, for every object and every isomorphism in , there exists an isomorphism in such that .
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.
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.