# nLab homotopical category

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## 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.

## References

This definition is in page 23 of

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

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

Last revised on December 19, 2021 at 18:17:15. See the history of this page for a list of all contributions to it.