I am reading lecture notes on topology and the total space $E(U(N))$ is given as a geometric realization of a simplicial space $$E(U(N))=|[n]\rightarrow U(N)^{n+1}|$$ Here I am confused because

1) the details about face and degeneracy maps are missing. What are they exactly?

2) I am confused that left handside depends on $n$ and right handside does not. Is it correct that I should have written $$E(U(N))=|[n]\rightarrow U(N)^{n+1}|$$ whatever that means. Probably, I wasn't careful to take the notes in the class. Wikipedia and ncatlab provide the simplicial space for the $BG$ and they give the special case for $EU$ geometrically without explaining the face/degeneracy maps.

  • $\begingroup$ which lecture notes are you referring to? can you give link? $\endgroup$ – Praphulla Koushik Jan 22 at 13:43
  • $\begingroup$ The right hand side has incorrect indexing $U(n)$ should be $U(N)$. You should look up the Bar construction. This is an example of one, the face and degeneracy maps will be explained in any reference on that, but the idea is that we are doing the face maps through multiplication in $U(N)$ and the degeneracy maps through inserting the identity. $\endgroup$ – Connor Malin Jan 22 at 13:48
  • $\begingroup$ @ConnorMalin thanks, could you advise any reference? $\endgroup$ – whoami Jan 22 at 13:49
  • $\begingroup$ @ConnorMalin that more or less solves my problem, because I couldn't get $n-1$ by $n-1$ unitary matrix from $n$ by $n$ unitary matrix. Deleting row and column didn't help but now I don't need it. $\endgroup$ – whoami Jan 22 at 13:52
  • $\begingroup$ people.math.binghamton.edu/malkiewich/bar.pdf $\endgroup$ – Connor Malin Jan 22 at 13:55

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