Infinite loop space

Summary

An infinite loop space is a space $X$ that is the loop space of a space. That space is again a loop space, and so on without end. Infinite loop spaces are exactly the zeroth spaces of $\Omega$-spectra, so they are the unstable shadows of spectra and of generalized cohomology theories. Recognition principles, such as the May recognition principle, detect them by a homotopy-coherent commutative multiplication.