Higher Groups, Spectra, and Stabilization

Prerequisites. Direct starred prerequisites: none. No later core chapter depends on this route.

Opening obstruction

A loop space carries multiplication and coherence, but repeated delooping imposes stronger stable structure than an arbitrary higher group possesses.

Development contract

Develop finite deloopings and higher groups around the stabilization unit ${\eta }_X:X\to \mathrm{\Omega }\mathrm{\Sigma }X$: for $n$-connected $X$, prove that ${\eta }_X$ is $2n$-connected and state the induced equivalences only below that truncation bound. Then identify coherently infinitely deloopable groups, rather than arbitrary higher groups, with connective spectra.