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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.
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.
This remains true in case of a monoidal model category under the additional assumption that the projection
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 (which following definition says a functor between categories is called part of the couniversal property of the homotopy category) is a quasi monoidal fibrations functor. 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 .
This implies that a enriched category gives rise to a -enriched category with the same objects as and the objects are defined by
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 model category. Let be denote an the category of small -enriched category. categories.
(1) Given A a morphism monoidal structure on in also itshomotopy category is called anequivalence if carries a monoidal structure which is determined up to a unique isomorphism by the homotopy requirement class that there exists a monoidal functor 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) from An -enriched functor to its homotopy category. 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 a monoidal model category.
A model functor category is in calledexcellent model category if it is equipped with a symmetric weak monoidal equivaleence structure if and satisfies the following induced conditions functor is an equivalence of -enriched categories.
(A1) In other words: F is a weak equivalence iff is combinatorial.
(A2) (1) Every For monomorphism every in pair , is a cofibration and the collection induced of map cofibrations in is stable under products.
(A3) The collection of weak equivalencies in is stable under filtered colimits.
(A4) is a weak equivalence in . is a Quillen bifunctor.
(A5) (2) The monoidal model category satisfies is essentially surjective on the invertibility level hypothesis. 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.