---
tag: 26GTNMAF7B
kind: concept
name: Stable homotopy category
msc: [55P42, 18G80]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_(topology)
---

## Summary

The stable homotopy category has [[spectra|1GHF4EKBXS]] as objects and homotopy classes of maps of spectra as morphisms.
In it the stable weak equivalences are made invertible.
In it the suspension functor is an equivalence, every morphism set is an abelian group, and cofibre sequences are also fibre sequences.
It is a triangulated category with a symmetric monoidal smash product, and it is the setting of [[stable homotopy theory|545EV0W4CE]].

## Explanations

### L1 human

Spaces can be suspended, but not desuspended.
In the stable homotopy category every object can also be desuspended, so shifting up and down are inverse operations.
Maps between objects form abelian groups, much as linear maps between vector spaces do.
For two finite CW complexes, the maps between their suspension spectra are exactly the stable homotopy classes of maps between them.
For example, the maps from the sphere spectrum to itself form the group $\Z$, given by degree.