homotopy theory, (∞,1)-category theory, homotopy type theory
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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.
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.)
See at free infinite loop space.
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
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