nLab
infinite loop space

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.

Related concepts

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

• Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. (euclid:1183538891, doi:10.1090/S0002-9904-1977-14318-8)

  Infinite loop space theory revisited (pdf)

• John Adams, Infinite loop spaces, Hermann Weyl lectures at IAS, Annals of Mathematics Studies 90, Princeton University Press (1978) (ISBN:9780691082066, doi:10.1515/9781400821259)

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

Section 5.1.3 of

• Jacob Lurie, Higher Algebra