May recognition principle

Remarks

May proved the principle in 1972 by building the spectrum from iterated bar constructions. The key input is his approximation theorem. It says that, for a connected space $Y$, the free algebra on $Y$ over the little $n$-cubes operad is weakly equivalent to $\Omega^n \Sigma^n Y$. Boardman and Vogt gave another proof with their theory of homotopy-everything spaces. The finite version holds too: a grouplike algebra over the little $n$-cubes operad is weakly equivalent to an $n$-fold loop space. If $X$ is not grouplike, the zeroth space of the spectrum is instead the group completion of $X$.

At work: the disjoint union $\coprod_{n \ge 0} B\Sigma_n$ is an algebra over the Barratt–Eccles operad, but $\pi_0 = \N$ is not a group. Its group completion is $\Omega^\infty \Sigma^\infty S^0 = QS^0$, which is the Barratt–Priddy–Quillen theorem.

Going deeper

For a student

The principle is an "infinite loop space machine": it turns $E_\infty$ spaces into spectra. It applies to classifying spaces of symmetric monoidal categories, which is how algebraic K-theory spectra are built. It also shows that spaces such as $BO$, $BU$ and $BTop$ are infinite loop spaces, which gives new cohomology theories and operations. Different machines, by May, Segal and Boardman–Vogt, give equivalent spectra by the uniqueness theorem of May and Thomason.