nLab infinite loop space



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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 with a choice of deloopings are equivalent to connective spectra.


Algebraic characterization

Infinite loop spaces with a choice of deloopings 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.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
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


Theorem of

Last revised on August 17, 2023 at 13:04:01. See the history of this page for a list of all contributions to it.