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I'm trying to read a paper by Hovey on model categories, so I turned to his book, Model Categories. Given a category, he gives the definition in terms certain morphisms called weak equivalences, fibrations and cofibrations.

A weak a equivalence is a morphism that becomes an iso when passed to the homotopy category. And I know the topological defintions of (co)fibrations.

Is there a categorical definition of fibration and cofibration? Or should I take them as undefined terms subject the axioms of a model category? I've even thought about constructing the category of CW-complexes, letting my weak equivalences be weak homotopies, and fibrations and cofibrations be the topological versions. Then try to see what the (co)fibrations do categorically speaking.