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Let be an object in a model category.
A cylinder object is defined to be a factorization of the codiagonal map for into a cofibration followed by a weak equivalence.
A path object is defined to be a factorization of the diagonal map for into a weak equivalence followed by a fibration .
Let be a model category. Let be a cofibrant object of . Let be a fibrant object of . Let be two parallel morphisms. Then the following conditions are equivalent.
The coproduct map factors through every cylinder object for .
The coproduct map factors through some cylinder object for .
The product map factors through every path object for .
The product map factors through some path object for .
(homotopy, homotopy category of a model category)
Let be a model category.
(1) Two maps from a cofibrant object to a fibrant object satisfying the conditions of Proposition A.2.2.1 are called homotopic morphisms. Homotopy is an equivalence relation on .
(2) The homotopy category of is defined to have as objects the fibrant-cofibrant objects of . The hom objects are defined to be the set of equivalence classes of .
The homotopy category (more precisely the projection map ) is couniversal in the following sense:
The second condition implies that the functor in the first condition is unique up to unique isomorphism.
The following remark gives an alternative equivalent definition of the homotopy category of a model category:
The homotopy category (more precisely the projection map ) is couniversal in the following sense:
The second condition implies that the functor in the first condition is unique up to unique isomorphism.
As always is the the case with (co)universal properties the object in question can be defined by this property.