If $X$ is grouplike, a classifying space $BX$ should be characterized by an $H$-map $X\to \Omega BX$ that is a weak equivalence. In the $E_2$-case, where one doesn't need to worry about non-commutativity, a two-fold classifying space $B^2X$ should be characterized by an $H$-map $X\to \Omega^2B^2X$ that is a group completion. I didn't study the non-connected case in Geo, but rectified that a bit later in ``$E_{\infty}$ spaces, group completions, and permutative categories''.