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This is a subentry of a reading guide to HTT.
A (This is Joyal’s definition; it differs from A.2.1.1 in that Joyal requestsmodel category is a category to be finitely bicomplete.)
A model category is a category equipped with three distinguished classes of -morphisms: The classes , , of cofibrations, fibrations, and weak equivalences, respectively, satisfying the following axioms:
admits (small) limits and colimits.
The class of weak equivalences satisfies 2-out-of-3.
and are weak factorization systems.
A model category is a category equipped with three distinguished classes of morphisms in : The classes , , of cofibrations, fibrations, and weak equivalences, respectively, satisfying the following axioms:
The classes and is closed under retracts. (by weak factorization systems, Lemma 2, in joyal’s catlab)
The class is closed under retracts. (by model categories, Lemma 1, in joyal’s catlab)
admits (small) limits and colimits.
The class of weak equivalences satisfies 2-out-of-3.
,and are closed under retracts.
and are weak factorization systems.
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 codiagonal map for into a weak equivalence followed by a fibration .