stable homotopy theory
loop space object
stable (∞,1)-category of spectra
stable homotopy category
smash product of spectra
symmetric smash product of spectra
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The sphere spectrum is the simplest nontrivial spectrum.
The sphere spectrum is the suspension spectrum of the point.
The sphere spectrum is naturally an E-∞ ring and in fact is the initial object in the (∞,1)-category of ring spectra. It is the higher version of the ring ℤ of integers.