---
date: 2021-10-21 18:42
modification date: Friday 22nd October 2021 23:42:13
title: homotopy category
aliases: [homotopy category, homotopy categories]

---

Tags: #todo  

# homotopy category

The homotopy category of $\cat{C}$: same objects, $\ho\cat{C}(x, y) := \pi_0 \cat{C}(x, y]$.

![Homotopy category](attachments/image_2021-03-25-00-45-13.png)

Define $\ho\cat C$ as the universal category equipped with a functor $\cat C \to \ho \cat C$ sending weak equivalences to isomorphisms.
Morphism
\[
\cat{C}(x, y) = \ho\cat{C}(RQx, RQy)
\]