In the Brown's book “Cohomology of groups”, chapter 5.1, there is a concept diagonal approximation, maybe that is not a standard definition, I feel something hard to understand it.
The book says that Alexander–Whitney map is a well-known example of the diagonal approximation, that is different with the diagonal map and I can not find some meaning of approximation. I want to know the motivation of it and show it in some examples. Who can help me?