Idea

A homotopical category is a construction used in homotopy theory, related to but more flexible than a model category.

Definition

A homotopical category is a category with weak equivalences where on top of the 2-out-of-3-property the morphisms satisfy the

2-out-of-6-property:

If morphisms $h \circ g$ and $g \circ f$ are weak equivalences, then so are $f$ , $g$ , $h$ and $h \circ g \circ f$ .

The 2-out-of-6-property implies the 2-out-of-3-property.
A functor $F : C \to D$ between homotopical categories which preserves weak equivalences is a homotopical functor.

Every homotopical category $C$ “presents” or “models” an (infinity,1)-category $L C$ , a simplicially enriched category called the simplicial localization of $C$ , which is in some sense the universal solution to inverting the weak equivalence up to higher categorical morphisms.

This definition is in paragraph 33 of

William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith. Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs.