Every model category is a homotopical category.
A functor between homotopical categories which preserves weak equivalences is a homotopical functor.
Every homotopical category “presents” or “models” an (infinity,1)-category , a simplicially enriched category called the simplicial localization of , which is in some sense the universal solution to inverting the weak equivalence up to higher categorical morphisms.
category with a calculus of fractions
This definition is in paragraph 33 of