TITLE: Intuition for the construction of the space $M_G=EG\times _G M$
QUESTION [5 upvotes]: Reference: Atiyah & Bott, The moment map and Equivariant cohomology
Question: What could be the motivation and the intuition behind the construction of the space $M_G=EG\times _G M$? When I am studying the equivariant cohomology, I have no idea about why we would like to define the equivariant cohomology as the ordinary cohomology of $M_G$.
A simple case is, if $G$ is a compact Lie group and acts freely on $M$, then $M_G$ is simply $M/G$, the space of all $G$-orbits.
But, in general cases, how to understand the picture of $M_G$? And, how does $M_G$ relate to $M/G$?
For example, take $G=S^1\times S^1$ and $M=\mathbb P^1$, and the action is given by $(t_0,t_1): [x,y]\mapsto [t_0x, t_1y]$. Then this action is not free and it has two fixed point $[1,0]$ and $[0,1]$. In this case, what is the difference between $M/G$ and $M_G$?

REPLY [6 votes]: In homotopy theory and in homological algebra, there a general idea that for any given operation that you might want to perform, there's a class of "nice" objects for that operation. It is a good idea to replace your object by another one which is "nice" and "equivalent" in some appropriate sense before performing the operation.
If the operation is $-/G$ for some group $G$, then the class of "nice" objects is $G$-spaces with a free action. It's a good idea to replace $M$ by $M\times EG$ before performing $-/G$.
If the operation is $\times_BX$ for some map $X\to B$, the the class of "nice" objects are the spaces equipped with a map to $B$ which is a fibration. 
If the operation is $-\otimes_RM$ for some ring $R$ and some $R$-module $M$, then the class of "nice" objects are the complexes of projective modules.
If the operation is $Hom_R(M,-)$  for some ring $R$ and some $R$-module $M$, then the class of "nice" objects are the complexes of injective modules.