nLab E-infinity space

Definition

An $E_\infty$-space is a commutative ∞-monoid in Top.

Properties

Grouplike $E_\infty$-spaces are equivalently infinite loop spaces. See there for more details.

Related concepts

• I-space

• connective spectrum

• model structure for dendroidal left fibrations

• ∞-group completion

• very commutative object

	(∞,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 with contributions by Frank Quinn, Nigel Ray and Jørgen Tornehave, $E_\infty$-Ring spaces and $E_\infty$ ring spectra, Lecture Notes in Mathematics 577, Springer 1977 (pdf, cds:1690879)