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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 .