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This is a subentry of HTT, A.3 and of a reading guide to HTT.
(Quillen bifunctor)
Let be model categories.
A functor is called Quillen bifunctor if the following conditions are satisfied:
(1) For cofibrations , and in resp. in , the induced map
is a cofibration in . Moreover is acyclic if either or is acyclic; where the pushout is
(2) preserves small colimits in each variable seperately.
setting shows that condition 1. in the previous definition reduces to the requirement on to preserve cofibrations and acyclic cofibrations.
(monoidal model category)
A monoidal model category is a monoidal category equipped with a model structure satisfying the following:
The tensor product is a left Quillen bifunctor.
The unit object is cofibrant.
The monoidal structure is closed.
The category is a monoidal model category with respect to the cartesian product and the Kan model structure.
(-enriched model category)
Let be a monoidal model category.
A -enriched model category is defined to be a -enriched category equiped with a model structure satisfying the following:
is tonsured and cotensored over .
The tensor product is a left Quillen bifunctor
(alternative characterization of the Quillen bifunctor )
Let , be -enriched model categories. Let be a Quillen adjunction between the underlying model categories. Let every object of be cofibrant. Let
be a weak equivalence for every pair of cofibrant objects , . Then the following are equivalent:
is a Quillen equivalence.
The restriction of determines a weak equivalence of -enriched categories .
The following remark explicates the relation between simplicial homotopy theory and model-category-theoretic homotopy theory.
Let be a Quillen equivalence between simplicial model categories category, where let every object of is be cofibrant. a Let cofibrant object of , be let a simplicial functor. Then induces be an a equivalence fibrant object of -categories . Then we have.
(1) is a Kan complex.
(2)
Let , be -enriched model categories. Let be a Quillen adjunction between the underlying model categories. Let every object of be cofibrant. Let
be a weak equivalence for every pair of cofibrant objects , . Then the following are equivalent:
is a Quillen equivalence.
The restriction of determines a weak equivalence of -enriched categories .
Let be a Quillen equivalence between simplicial model categories where every object of is cofibrant. Let be a simplicial functor. Then induces an equivalence of -categories .