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, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113, AMS 2004 (ISBN: 978-1-4704-1340-8, pdf)

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

Survey with an eye towards (∞,1)-categories:

• Emily Riehl, Homotopical categories: from model categories to (∞,1)-categories, (arXiv:1904.00886) in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds,) Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press