nLab
homotopical category

Idea

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

Definition

A homotopical category is a category with a distinguished class of morphisms (called ‘weak equivalences’) satisfying the following conditions:

• Every identity map is a weak equivalence.

• It has 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 $h \circ g \circ f$.

Remarks

• If the first condition in Def. 2 holds, then the 2-out-of-6-property implies the 2-out-of-3 property, hence every homotopical category is a category with weak equivalences.

• Every model category yields a homotopical category.

• A functor $F : C \to D$ between homotopical categories which preserves weak equivalences is a homotopical functor.

Simplicial localization

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.

Related concepts

• relative category

• category of fibrant objects

• cofibration category

• Waldhausen category

• model category

• category with a calculus of fractions

• resolution

References

This definition is in page 23 of

• William Dwyer, Philip Hirschhorn, Daniel Kan and Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs. (pdf)

with the main development of the concept starting in subsection 33 on page 96.