Contents

# Contents

## Idea

A topological space is a loop space if it has a delooping. It is an infinite loop space if this delooping has itself a delooping, and so on.

In homotopy theory infinite loop spaces are equivalent to connective spectra.

## Properties

### Algebraic characterization

Infinite loop spaces are the grouplike E-∞ algebras in Top (grouplike E-∞ spaces).

See for instance (Adams, pretheorem 2.3.2) and the references listed there for traditional accounts. See (Lurie, section 5.1.3) for a modern formulation.

(Compare to how just loop spaces are the grouplike A-∞ algebras, see looping and delooping.)

### Free infinite loop space

See at free infinite loop space.

E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ ∞-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object