Let $G$ be a discrete group with universal principal bundle $EG\to BG$, and let $X$ and $Y$ be left $G$-spaces. An equivariant map $\overline{f}:X\to Y$ induces a fibre-preserving map $f:EG\times\_G X\to EG\times\_G Y$ between Borel constructions, as in the following diagram.


$\require{AMScd}$
\begin{CD}
 X @>\overline{f}>> Y\\
 @V V V @VV V\\
 EG\times\_G X @>>f> EG\times\_G Y \\
 @V V V @VV V\\
 BG @>=>> BG
\end{CD}


I believe some sort of converse to be true. That is, if $f:EG\times\_G X\to EG\times\_G Y$ is a fibre-preserving map between Borel constructions, it induces a $G$-equivariant map $\overline{f}:X\to Y$. Here I am identifying $G=\pi\_1(BG,\ast)$, which acts (up to homotopy) on the fibres.



> 
> Does anyone know of a reference to a precise statement along these lines?
> 
> 
> 


**Edit** (in response to user51223's comment): I'm mainly interested in the existence question: Is it true that there exists an equivariant map $\overline{f}:X\to Y$ if and only if there exists a map $f:EG\times\_G X\to EG\times\_G Y$ over $BG$? (So $\overline{f}$ doesn't have to be the induced map on fibres.) 



Fibre preserving maps of Borel constructions
The answer to the question in the edit is no. Take $G=\mathbb Z$, $X$ to be a point and $Y=\mathbb R$ with the action $n\cdot x = x+n$. Then there are no equivariant maps from $X$ to $Y$ but there are many maps from $EG \times\_G X = BG = S^1$ to the cylinder $EG \times\_G Y$ over $BG$.


In this special case when $X$ is a point, an equivariant map $X \to Y$ is a fixed point, while a map $EG \times\_G X \to EG \times\_G Y$ is a homotopy fixed point. The fixed points include in the homotopy fixed points, but in general I don't think there is much one can say about the inclusion.


