symmetric monoidal (∞,1)-category of spectra
that means it is equipped with
a binary product operation
a choice of pentagon law? homotopy between five such s;
and so ever on, as controled by the associahedra.
If in the definition of an -space one discards all the higher homotopies and retains only the existence of an associativity-homotopy, then one has the notion of H-monoid. Put another way, An -space in the (∞,1)-category ∞Grpd/Top becomes an H-monoid in the homotopy Ho(Top). And lifting an H-monoid structure to an -space structure means lifting a monoid structure through the projection from the (∞,1)-category ∞Grpd/Top to Ho(Top).
The delooping of an -space is an A-∞ category/(∞,1)-category with a single object. (Beware that in standard literature ”-category” is often, but not necessarily, reserved for a stable (∞,1)-category).
|(∞,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 Γ-space||infinite loop space object|
|connective spectrum||connective spectrum object|