---
tag: A7VB2NSGJ2
kind: concept
name: Infinite loop space
msc: [55P47, 55P42]
sources:
  - url: https://en.wikipedia.org/wiki/Infinite_loop_space_machine
---

## Summary

An infinite loop space is a space $X$ that is the [[loop space|2JFF5F3EGF]] of a space.
That space is again a loop space, and so on without end.
Infinite loop spaces are exactly the zeroth spaces of $\Omega$-spectra, so they are the unstable shadows of [[spectra|1GHF4EKBXS]] and of generalized cohomology theories.
Recognition principles, such as the [[May recognition principle|5SMK5KEBQE]], detect them by a homotopy-coherent commutative multiplication.

## Explanations

### L1 human

Loops in a space can be composed, so a loop space has a multiplication that is associative up to homotopy.
If the space is a double loop space, the multiplication is also commutative up to homotopy.
An infinite loop space has infinitely many such layers of structure, which make its multiplication as commutative as homotopy theory allows.
For an abelian group $A$, the Eilenberg–MacLane spaces $K(A, n)$ satisfy $K(A, n) \simeq \Omega K(A, n+1)$, so each of them is an infinite loop space.