Let $X$ a simplicial set, $x\in X_m$ an $m$-simplex. We say that $x$ is degenerate if $\exists s:[m]\to[n]$ an epimorphism such that $n<m$ and $y\in X_n$ such that $X(s)(y)=x$.

Now I want to show Eilenberg-Zilber's lemma, that is : for each $x\in X_m$, $\exists s:[m]\to [n]$ an epimorphism and $y\in X_n$ non degenerate $n$-simplex such that $X(s)(y)=x$, and $(s,y)$ is unique.

The lectures I found or books (Calculus of fractions and homotopy theory, p.26/27) say that the existence is obvious, and then proceeds to show uniqueness. Obvious, sure. I thought about taking $n$ minimal in some sense but which sense ? And how to get this $y$? How can we be sure that it's non degenerate ?

In fact my question is simple, how is existence shown ?

  • 1
    $\begingroup$ This is proven in the Kerodon. Calculus of Fractions and Homotopy Theory is quite hard to read, not the best of books in that regard $\endgroup$
    – FShrike
    Mar 27 at 10:07
  • $\begingroup$ I didn't know this thank you, however I went on the website and checked a 1600 pages notes but I didn't find it by keyword, is it the same result or some different version ? $\endgroup$
    – raisinsec
    Mar 27 at 10:17
  • 1
    $\begingroup$ Proposition kerodon.net/tag/0010 $\endgroup$
    – FShrike
    Mar 27 at 11:00


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