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.

A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ ∞-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

## References

• Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. (Euclid)

Infinite loop space theory revisited (pdf)

• John Adams, Infinite loop spaces, Hermann Weyl lectures at IAS, Princeton University Press (1978)

• Peter May, The uniqueness of infinite loop space machines, Topology, vol 17, pp. 205-224 (1978) (pdf)

Section 5.1.3 of

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