Let $G\_1$ and $G\_2$ be topological groups. Assume that there exists a continuous homomorphism $f : G\_1 \rightarrow G\_2$ which (ignoring the group structure) is a homotopy equivalence. If $BG\_i$ is a classifying space for $G\_i$, then we get an induced map $f\_{\ast} : BG\_1 \rightarrow BG\_2$ which is a homotopy equivalence.


The fact that $f\_{\ast}$ is a homotopy equivalence is clear if you construct $BG\_i$ via Milnor's construction of classifying spaces. However, the existence of $BG\_i$ is also an easy consequence of the (unpointed) Brown representability theorem. This leads me to my question : is there an "abstract nonsense" proof that $f\_{\ast}$ is a homotopy equivalence which avoids having to know an explicit description of $BG\_i$?


EDIT : As Fernando Muro pointed out, the desired result is equivalent to the fact that if $EG \rightarrow BG$ is the universal $G$-bundle over $BG$, then the space $EG$ is contractible. Of course, this is true, but I don't see how to prove it without knowing an explicit construction of $BG$. My definition of $BG$ and the universal bundle $EG \rightarrow BG$ is that $BG$ is a space such that for any CW complex $X$, the set of principal $G$-bundles on $X$ is naturally in bijection with $[X,BG]$ via the map that takes $\phi : X \rightarrow BG$ to $\phi^{\ast}(EG)$. Is there a way to see that $EG$ has to be contractible without knowing a construction of $BG$?



Classifying spaces, Brown representability, and homotopy equivalences
The space $\Omega BG$ is, by categorical nonsense, a classifying space for principal $G$-bundles over $\Sigma X$ with a chosen trivialisation at the basepoint. Any such bundle can be trivialised over the upper and lower halves of $\Sigma X$, and the difference between the two trivialisations gives a map $X\to G$. One can show using this that $G$ represents the same functor as $\Omega BG$, so they are homotopy equivalent. (We probably want to assume here that $G$ has the homotopy type of a CW complex). In fact, this line of argument gives a map $G\to\Omega BG$ that is well-defined and natural up to homotopy. This should be enough to show that $\Omega(f\_\*)$ is a homotopy equivalence, and thus that $\pi\_\*(f\_\*)$ is an isomorphism, so $f\_\*$ is a weak equivalence.


