---
tag: 9Y5Y92R4D5
kind: concept
name: Homotopy category of chain complexes
msc: [18G80, 18G35]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_category_of_chain_complexes
same_as:
  wikidata: Q1430845
  wikipedia: Homotopy category of chain complexes
---

## Summary

Let $\mathcal{A}$ be an additive category.
The homotopy category of chain complexes $K(\mathcal{A})$ has the [[chain complexes|WQ82C0M3QB]] as objects and the [[chain maps|HTK9WQZJ1G]] modulo [[chain homotopy|4EBHAWJJ8C]] as morphisms.
In it, chain homotopy equivalences become isomorphisms.
It is a [[triangulated category|T2V7GJJ2EN]], and the [[derived category|5ST7NVJFSM]] of an abelian category is obtained from it by inverting the quasi-isomorphisms.

## Explanations

### L1 human

Two chain maps that are chain homotopic induce the same map on homology, so for many purposes they are the same.
The homotopy category identifies them.
A complex that can be contracted to zero, such as $0 \to \Z \xrightarrow{\id} \Z \to 0$, becomes isomorphic to the zero complex.
It is the algebraic analogue of the homotopy category of spaces, where homotopic maps are identified.