I am thinking about the Eilenberg-MacLane spaces $K(\mathbb{Z}, 1)$ and $K(\mathbb{Z}, 2)$.
I understand the usual construction of these spaces involves $EG$ and $BG$, $G = \mathbb{Z}$. $EG$ is a homotopy colimit of $*$ with $\mathbb{Z}$ as a diagram (a group), and $BG$ is the quotient. We obtain $U(1) = S^1$ in a rather canonical way.
I was wondering if there is another way to construct $BG$ - entirely in terms of common homotopical constructions - but which produces $\mathbb{C}^\times$ instead.
This is of interest to me since I am interested in how more rigid structures like Riemann surfaces are related to purely homotopical constructions. For that reason, it would be great to know about how e.g. the multiplication on $\mathbb{C}^\times = BG$ or the addition on $\mathbb{C} = EG$ could arise in terms of homotopy colimits or something homotopically natural.