Principal G Bundles and Lifts of Maps of the Base Space

Question

I'm reading this paper and need a little helping understanding example 1.4. In particular, the author writes

To see this note that $f$ can be lifted to a map $\tilde f: EG \to EH$, such that the following diagram commutes

Please see the paper for the diagram. The author then writes

and $\tilde f$ satisfies $\tilde f(eg)=\tilde f(e)\alpha(g)$.

So my questions are

1) What is the lifting property he is using? The only one I'm used to is the one from Hatcher's book on algebraic topology. Is he using some sort of generalization of that?

2) How does he know that $\tilde f$ is $G$-eqivariant. I.e., $\tilde f(eg)=\tilde f(e)\alpha(g)$.

Answer 1

You can use the Milnor join to define $\tilde f$ directly. Milnor as represented the $EG$ space as the limit of spaces $(g_1,t_1,...,g_n,t_n)$ with $\sum t_i=1$ quotiented by a relation. (see the reference). $G$ acts naturally on this space and the quotient is $BG$. Every map $\alpha:G\rightarrow H$ induces naturally a map $EG\rightarrow EH$ which satisfies the properties that you have mentioned.

http://www.jstor.org/stable/1970012?seq=1#page_scan_tab_contents