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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?

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    I am not sure what do you need. But you can view it as a simplicial approximation to diagonal map. This was the suggested motivation for it in the lecture notes by J.F.Davis.2015-04-02

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