# nLab homotopical category

## Theorems

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# 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 $h \circ g \circ f$.

# 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.

# References

This definition is in paragraph 33 of

Revised on January 3, 2012 05:26:48 by Mike Shulman (173.8.161.189)